![Page 1: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/1.jpg)
CSE 311 Lecture 10: Set TheoryEmina Torlak and Kevin Zatloukal
1
![Page 2: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/2.jpg)
TopicsEnglish proofs and proof strategies
A quick wrap-up of .Set theory basics
Set membership ( ), subset ( ), and equality ( ).Set operations
Set operations and their relation to Boolean algebra.More sets
Power set, Cartesian product, and Russell’s paradox.Working with sets
Representing sets as bitvectors and applications of bitvectors.
Lecture 09
∈ ⊆ =
2
![Page 3: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/3.jpg)
English proofs and proof strategiesA quick wrap-up of .Lecture 09
![Page 5: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/5.jpg)
Benefits of English proofs
This is more work to write
than this
Higher level language is easierbecause it skips details.
%a = add %i, 1%b = mod %a, %n%c = add %arr, %b%d = load %c%e = add %arr, %istore %e, %d
arr[i] = arr[(i+1) % n];
4
![Page 6: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/6.jpg)
Benefits of English proofs
This is more work to write
than this
Higher level language is easierbecause it skips details.
Formal proofs are the low levellanguage: each part must be spelledout in precise detail.
%a = add %i, 1%b = mod %a, %n%c = add %arr, %b%d = load %c%e = add %arr, %istore %e, %d
arr[i] = arr[(i+1) % n];
4
![Page 7: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/7.jpg)
Benefits of English proofs
This is more work to write
than this
Higher level language is easierbecause it skips details.
Formal proofs are the low levellanguage: each part must be spelledout in precise detail.
English proofs are the high levellanguage.
An English proof is correct if thereader is convinced they can“compile” it to a formal proof ifnecessary.
%a = add %i, 1%b = mod %a, %n%c = add %arr, %b%d = load %c%e = add %arr, %istore %e, %d
arr[i] = arr[(i+1) % n];
4
![Page 8: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/8.jpg)
Proof strategiesSometimes, it’s too hard to prove a theorem directly using inferencerules, equivelances, and domain properties.
When that’s the case, try one of the following alternative strategies:
Proof by contrapositive,Disproof by counterexamples, andProof by contradiction.
5
![Page 9: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/9.jpg)
Proof by contrapositiveIf we assume and derive , then we have proven that ,which is equivalent to proving .
1.1. Assumption 1.3.
2. Direct Proof Rule3. Contrapositive: 2
¬q ¬p ¬q → ¬pp → q
¬q…¬p
¬q → ¬pp → q
6
![Page 10: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/10.jpg)
Counterexamples
To disprove , prove .Works by DeMorgan’s Law: .All we need to do is find an for which is false.This is called a counterexample.
Example: disprove that “Every prime number is odd”.2 is a prime number that is not odd.
∀x. P(x) ∃x. ¬ P(x)¬∀x. P(x) ≡ ∃x. ¬P(x)x P(x)
x
7
![Page 11: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/11.jpg)
Proof by contradictionIf we assume and derive (a contradiction), then we have proven .
1.1. Assumption 1.3.
2. Direct Proof Rule3. Law of Implication: 24. Identity: 3
p ' ¬pp…'
p → '¬p ∨ '¬p
8
![Page 12: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/12.jpg)
An example proof by contradiction
Prove that “No integer is both even and odd.”English proof: .
Proof by contradiction
¬∃x. Even(x) ∧ Odd(x) ≡ ∀x. ¬(Even(x) ∧ Odd(x))
9
![Page 13: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/13.jpg)
An example proof by contradiction
Prove that “No integer is both even and odd.”English proof: .
Proof by contradictionLet be an arbitrary integer and suppose that it is both even and odd.
¬∃x. Even(x) ∧ Odd(x) ≡ ∀x. ¬(Even(x) ∧ Odd(x))
x
9
![Page 14: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/14.jpg)
An example proof by contradiction
Prove that “No integer is both even and odd.”English proof: .
Proof by contradictionLet be an arbitrary integer and suppose that it is both even and odd.Then for some integer and for some integer .
¬∃x. Even(x) ∧ Odd(x) ≡ ∀x. ¬(Even(x) ∧ Odd(x))
xx = 2a a x = 2b + 1 b
9
![Page 15: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/15.jpg)
An example proof by contradiction
Prove that “No integer is both even and odd.”English proof: .
Proof by contradictionLet be an arbitrary integer and suppose that it is both even and odd.Then for some integer and for some integer .Therefore and hence .
¬∃x. Even(x) ∧ Odd(x) ≡ ∀x. ¬(Even(x) ∧ Odd(x))
xx = 2a a x = 2b + 1 b
2a = 2b + 1 a = b + 12
9
![Page 16: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/16.jpg)
An example proof by contradiction
Prove that “No integer is both even and odd.”English proof: .
Proof by contradictionLet be an arbitrary integer and suppose that it is both even and odd.Then for some integer and for some integer .Therefore and hence .But two integers cannot differ by so this is a contradiction.
¬∃x. Even(x) ∧ Odd(x) ≡ ∀x. ¬(Even(x) ∧ Odd(x))
xx = 2a a x = 2b + 1 b
2a = 2b + 1 a = b + 121
2
9
![Page 17: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/17.jpg)
An example proof by contradiction
Prove that “No integer is both even and odd.”English proof: .
Proof by contradictionLet be an arbitrary integer and suppose that it is both even and odd.Then for some integer and for some integer .Therefore and hence .But two integers cannot differ by so this is a contradiction.Therefore no integer is both even and odd.
¬∃x. Even(x) ∧ Odd(x) ≡ ∀x. ¬(Even(x) ∧ Odd(x))
xx = 2a a x = 2b + 1 b
2a = 2b + 1 a = b + 121
2◻
9
![Page 18: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/18.jpg)
Fun strategy: proof by computer an automated theorem prover:Use
; No integer is both even and odd.
(define-fun even ((x Int)) Bool (exists ((y Int)) (= x (* 2 y))))
(define-fun odd ((x Int)) Bool (exists ((y Int)) (= x (+ (* 2 y) 1))))
(define-fun claim () Bool (not (exists ((x Int)) (and (even x) (odd x)))))
(assert (not claim)) ; proof by contradiction
(check-sat)
10
![Page 19: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/19.jpg)
Fun strategy: proof by computer an automated theorem prover:
While this example works, proofs of arbitrary formulas in predicate logiccannot be automated. But interactive theorem provers can still help bychecking your formal proof and filling in some low-level details for you.
Use
; No integer is both even and odd.
(define-fun even ((x Int)) Bool (exists ((y Int)) (= x (* 2 y))))
(define-fun odd ((x Int)) Bool (exists ((y Int)) (= x (+ (* 2 y) 1))))
(define-fun claim () Bool (not (exists ((x Int)) (and (even x) (odd x)))))
(assert (not claim)) ; proof by contradiction
(check-sat)
10
![Page 20: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/20.jpg)
Fun fact: counterexamples & contradiction in verification
Automated verifiers work by counterexample and contradiction proofs.Recall that program verification involves proving that a program satisfies a specification on all inputs : , where and are formulas encoding the semantics of and .
PS x ∀x. p(x) → s(x) p s
P S
11
![Page 21: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/21.jpg)
Fun fact: counterexamples & contradiction in verification
Automated verifiers work by counterexample and contradiction proofs.Recall that program verification involves proving that a program satisfies a specification on all inputs : , where and are formulas encoding the semantics of and .
The program verifier sends the formula to the prover..
PS x ∀x. p(x) → s(x) p s
P S∃x. p(x) ∧ ¬ s(x)
¬∀x. p(x) → s(x) ≡ ∃x. ¬(p(x) → s(x)) ≡ ∃x. ¬(¬p(x) ∨ s(x)) ≡ ∃x. p(x) ∧ ¬s(x)
11
![Page 22: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/22.jpg)
Fun fact: counterexamples & contradiction in verification
Automated verifiers work by counterexample and contradiction proofs.Recall that program verification involves proving that a program satisfies a specification on all inputs : , where and are formulas encoding the semantics of and .
The program verifier sends the formula to the prover..
If the prover finds a counterexample, we know the program is incorrect.The counterexample is a concrete input (test case) on which the programviolates the spec.
PS x ∀x. p(x) → s(x) p s
P S∃x. p(x) ∧ ¬ s(x)
¬∀x. p(x) → s(x) ≡ ∃x. ¬(p(x) → s(x)) ≡ ∃x. ¬(¬p(x) ∨ s(x)) ≡ ∃x. p(x) ∧ ¬s(x)
11
![Page 23: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/23.jpg)
Fun fact: counterexamples & contradiction in verification
Automated verifiers work by counterexample and contradiction proofs.Recall that program verification involves proving that a program satisfies a specification on all inputs : , where and are formulas encoding the semantics of and .
The program verifier sends the formula to the prover..
If the prover finds a counterexample, we know the program is incorrect.The counterexample is a concrete input (test case) on which the programviolates the spec.
If no counterexample exists, we know the program is correct.Because this is proof by contradiction! The prover assumed
and arrived at false (“unsat”).
PS x ∀x. p(x) → s(x) p s
P S∃x. p(x) ∧ ¬ s(x)
¬∀x. p(x) → s(x) ≡ ∃x. ¬(p(x) → s(x)) ≡ ∃x. ¬(¬p(x) ∨ s(x)) ≡ ∃x. p(x) ∧ ¬s(x)
∃x. p(x) ∧ ¬s(x)
11
![Page 24: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/24.jpg)
Set theory basicsSet membership ( ), subset ( ), and equality ( ).∈ ⊆ =
12
![Page 25: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/25.jpg)
What is a set?
A set is a collection of objects called elements.Write to say that is an element in the set .Write to say that isn’t an element of .
Examples
a ∈ B a Ba ∉ B a B
A = {1}B = {1, 3, 2}C = {△, 1}D = {{17}, 17}E = {1, 2, 7, α, ∅, dog, }
13
![Page 26: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/26.jpg)
Some common sets
is the set of Natural Numbers: is the set of Integers: is the set of Rational Numbers: e.g. is the set of Real Numbers: e.g.
is the set where is a natural number. is the empty set; the only set with no elements.
ℕ ℕ = {0, 1, 2, …}ℤ ℤ = {… , −2, −1, 0, 1, 2, …}ℚ , −17,1
23248
ℝ 1, −17, , π,3248 2‾√
[n] {1, 2, … , n} n{} = ∅
14
![Page 27: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/27.jpg)
Sets can be elements of other sets
For example, consider the sets
Then we have
A = {{1}, {2}, {1, 2}, ∅}B = {1, 2}
B ∈ A∅ ∈ A
15
![Page 28: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/28.jpg)
Definitions: equality and subset
and are equal if they have the same elements.
is a subset of if every element of is also in .
A BA = B ≡ ∀x. x ∈ A ↔ x ∈ B
A B A BA ⊆ B ≡ ∀x. x ∈ A → x ∈ B
Note: .A = B ≡ A ⊆ B ∧ B ⊆ A
16
![Page 29: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/29.jpg)
Example: understanding equality
and are equal if they have the same elements.
Which sets are equal to each other?
A BA = B ≡ ∀x. x ∈ A ↔ x ∈ B
A = {1, 2, 3}B = {3, 4, 5}C = {3, 4}D = {4, 3, 3}E = {3, 4, 3}F = {4, {3}}
17
![Page 30: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/30.jpg)
Example: understanding equality
and are equal if they have the same elements.
Which sets are equal to each other?
are equal to each other.Order of elements doesn’t matter, and duplicates don’t matter.
is not equal to because !
A BA = B ≡ ∀x. x ∈ A ↔ x ∈ B
A = {1, 2, 3}B = {3, 4, 5}C = {3, 4}D = {4, 3, 3}E = {3, 4, 3}F = {4, {3}}
C, D, E
F C, D, E {3} ≠3
17
![Page 31: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/31.jpg)
Example: understanding subsets
is a subset of if every element of is also in .
Example sets
Are these subset formulas true or false?
A B A BA ⊆ B ≡ ∀x. x ∈ A → x ∈ B
A = {1, 2, 3}B = {3, 4, 5}C = {3, 4}
∅ ⊆ AA ⊆ BC ⊆ B
18
![Page 32: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/32.jpg)
Example: understanding subsets
is a subset of if every element of is also in .
Example sets
Are these subset formulas true or false?
A B A BA ⊆ B ≡ ∀x. x ∈ A → x ∈ B
A = {1, 2, 3}B = {3, 4, 5}C = {3, 4}
∅ ⊆ A 6A ⊆ BC ⊆ B
18
![Page 33: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/33.jpg)
Example: understanding subsets
is a subset of if every element of is also in .
Example sets
Are these subset formulas true or false?
A B A BA ⊆ B ≡ ∀x. x ∈ A → x ∈ B
A = {1, 2, 3}B = {3, 4, 5}C = {3, 4}
∅ ⊆ A 6A ⊆ B 'C ⊆ B
18
![Page 34: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/34.jpg)
Example: understanding subsets
is a subset of if every element of is also in .
Example sets
Are these subset formulas true or false?
A B A BA ⊆ B ≡ ∀x. x ∈ A → x ∈ B
A = {1, 2, 3}B = {3, 4, 5}C = {3, 4}
∅ ⊆ A 6A ⊆ B 'C ⊆ B 6
18
![Page 35: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/35.jpg)
Building sets from predicates
S = {x : P(x)}
is the set of all in the domainof for which is true.The domain of is o"en calledthe universe .
S xP P(x)
PU
19
![Page 36: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/36.jpg)
Building sets from predicates
S = {x : P(x)}
is the set of all in the domainof for which is true.The domain of is o"en calledthe universe .
S xP P(x)
PU
S = {x ∈ A : P(x)}
is the set of all in for which is true.
S x AP(x)
19
![Page 37: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/37.jpg)
Set operationsSet operations and their relation to Boolean algebra.
20
![Page 38: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/38.jpg)
Union, intersection, and set difference
Union
A B
Intersection
A B
Set difference
A B
A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
A ∖ B = {x : (x ∈ A) ∧ (x ∉ B)}
21
![Page 39: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/39.jpg)
Union, intersection, and set difference
Union
A B
Intersection
A B
Set difference
A B
A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
A ∖ B = {x : (x ∈ A) ∧ (x ∉ B)}
Given the following sets …
Use set operations to make:
A = {1, 2, 3}B = {3, 5, 6}C = {3, 4}
[6] ={3} ={1, 2} =
21
![Page 40: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/40.jpg)
Union, intersection, and set difference
Union
A B
Intersection
A B
Set difference
A B
A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
A ∖ B = {x : (x ∈ A) ∧ (x ∉ B)}
Given the following sets …
Use set operations to make:
A = {1, 2, 3}B = {3, 5, 6}C = {3, 4}
[6] = A ∪ B ∪ C{3} ={1, 2} =
21
![Page 41: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/41.jpg)
Union, intersection, and set difference
Union
A B
Intersection
A B
Set difference
A B
A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
A ∖ B = {x : (x ∈ A) ∧ (x ∉ B)}
Given the following sets …
Use set operations to make:
A = {1, 2, 3}B = {3, 5, 6}C = {3, 4}
[6] = A ∪ B ∪ C{3} = A ∩ B = A ∩ C{1, 2} =
21
![Page 42: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/42.jpg)
Union, intersection, and set difference
Union
A B
Intersection
A B
Set difference
A B
A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
A ∖ B = {x : (x ∈ A) ∧ (x ∉ B)}
Given the following sets …
Use set operations to make:
A = {1, 2, 3}B = {3, 5, 6}C = {3, 4}
[6] = A ∪ B ∪ C{3} = A ∩ B = A ∩ C{1, 2} = A ∖ B = A ∖ C
21
![Page 43: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/43.jpg)
Symmetric difference and complement
Symmetric difference
A B
Complement (with respect to universe )
A U
A ⊕ B = {x : (x ∈ A) ⊕ (x ∈ B)}
U= {x : x ∉ A} = {x : ¬(x ∈ A)}A
⎯ ⎯⎯⎯
22
![Page 44: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/44.jpg)
Symmetric difference and complement
Symmetric difference
A B
Complement (with respect to universe )
A U
A ⊕ B = {x : (x ∈ A) ⊕ (x ∈ B)}
U= {x : x ∉ A} = {x : ¬(x ∈ A)}A
⎯ ⎯⎯⎯
Given the sets and universe …
What is
A = {1, 2, 3}B = {1, 2, 4, 6}U = {1, 2, 3, 4, 5, 6}
A ⊕ B ==A
⎯ ⎯⎯⎯
22
![Page 45: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/45.jpg)
Symmetric difference and complement
Symmetric difference
A B
Complement (with respect to universe )
A U
A ⊕ B = {x : (x ∈ A) ⊕ (x ∈ B)}
U= {x : x ∉ A} = {x : ¬(x ∈ A)}A
⎯ ⎯⎯⎯
Given the sets and universe …
What is
A = {1, 2, 3}B = {1, 2, 4, 6}U = {1, 2, 3, 4, 5, 6}
A ⊕ B = {3, 4, 6}=A
⎯ ⎯⎯⎯
22
![Page 46: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/46.jpg)
Symmetric difference and complement
Symmetric difference
A B
Complement (with respect to universe )
A U
A ⊕ B = {x : (x ∈ A) ⊕ (x ∈ B)}
U= {x : x ∉ A} = {x : ¬(x ∈ A)}A
⎯ ⎯⎯⎯
Given the sets and universe …
What is
A = {1, 2, 3}B = {1, 2, 4, 6}U = {1, 2, 3, 4, 5, 6}
A ⊕ B = {3, 4, 6}=A
⎯ ⎯⎯⎯ {4, 5, 6}
22
![Page 47: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/47.jpg)
This is Boolean algebra again
Union is defined using .
Intersection is defined using .
Complement works like .
∪ ∨A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
∩ ∧A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
¬= {x : x ∉ A} = {x : ¬(x ∈ A)}A
⎯ ⎯⎯⎯
23
![Page 48: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/48.jpg)
This is Boolean algebra again
Union is defined using .
Intersection is defined using .
Complement works like .
∪ ∨A ∪ B = {x : (x ∈ A) ∨ (x ∈ B)}
∩ ∧A ∩ B = {x : (x ∈ A) ∧ (x ∈ B)}
¬= {x : x ∉ A} = {x : ¬(x ∈ A)}A
⎯ ⎯⎯⎯
This means that all equivalences from Boolean algebra translate directlyinto set theory, and you can use them in your proofs!
23
![Page 49: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/49.jpg)
DeMorgan’s laws
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
24
![Page 50: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/50.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
24
![Page 51: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/51.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
24
![Page 52: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/52.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
24
![Page 53: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/53.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)
24
![Page 54: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/54.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
24
![Page 55: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/55.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .Applying DeMorgan’s laws, we get .
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
x ∉ A ∧ x ∉ B
24
![Page 56: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/56.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .Applying DeMorgan’s laws, we get .So, by definition of complement.
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
x ∉ A ∧ x ∉ Bx ∈ ∧ x ∈A
⎯ ⎯⎯⎯B⎯ ⎯⎯⎯
24
![Page 57: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/57.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .Applying DeMorgan’s laws, we get .So, by definition of complement.Finally, by definition of “ ”, and wehave shown that .
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
x ∉ A ∧ x ∉ Bx ∈ ∧ x ∈A
⎯ ⎯⎯⎯B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯ ∩x ∈ → x ∈ ∩A ∪ B
⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
24
![Page 58: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/58.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .Applying DeMorgan’s laws, we get .So, by definition of complement.Finally, by definition of “ ”, and wehave shown that . Next, let be arbitrary.
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
x ∉ A ∧ x ∉ Bx ∈ ∧ x ∈A
⎯ ⎯⎯⎯B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯ ∩x ∈ → x ∈ ∩A ∪ B
⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
24
![Page 59: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/59.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .Applying DeMorgan’s laws, we get .So, by definition of complement.Finally, by definition of “ ”, and wehave shown that . Next, let be arbitrary.
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
x ∉ A ∧ x ∉ Bx ∈ ∧ x ∈A
⎯ ⎯⎯⎯B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯ ∩x ∈ → x ∈ ∩A ∪ B
⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
…
24
![Page 60: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/60.jpg)
DeMorgan’s laws
Proof technique:To prove , show
and.
Proof that :Let be arbitrary.Then, by definition of complement, .By definition of “ ”, .Applying DeMorgan’s laws, we get .So, by definition of complement.Finally, by definition of “ ”, and wehave shown that . Next, let be arbitrary.
Finally, , so ,which completes the proof.
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
= ∪A ∩ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
How would we prove these?
C = Dx ∈ C → x ∈ Dx ∈ D → x ∈ C
= ∩A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ A⎯ ⎯⎯⎯ B⎯ ⎯⎯⎯
x ∈ A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
¬(x ∈ A ∪ B)∪ ¬(x ∈ A ∨ x ∈ B)
x ∉ A ∧ x ∉ Bx ∈ ∧ x ∈A
⎯ ⎯⎯⎯B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯ ∩x ∈ → x ∈ ∩A ∪ B
⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
x ∈ ∩A⎯ ⎯⎯⎯
B⎯ ⎯⎯⎯
…x ∈ A ∪ B
⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯x ∈ ∩ → x ∈A
⎯ ⎯⎯⎯B⎯ ⎯⎯⎯
A ∪ B⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
◻
24
![Page 61: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/61.jpg)
Distributivity laws
A B
C
A B
C
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
25
![Page 62: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/62.jpg)
A simple set proof
Prove that for any sets and , we have .Recall that
Proof:
A B (A ∩ B) ⊆ AX ⊆ Y ≡ ∀x. x ∈ X → x ∈ Y
26
![Page 63: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/63.jpg)
A simple set proof
Prove that for any sets and , we have .Recall that
Proof:Let and be arbitrary sets and an arbitrary element of .
A B (A ∩ B) ⊆ AX ⊆ Y ≡ ∀x. x ∈ X → x ∈ Y
A B x A ∩ B
26
![Page 64: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/64.jpg)
A simple set proof
Prove that for any sets and , we have .Recall that
Proof:Let and be arbitrary sets and an arbitrary element of .Then, by definition of , we have that and .
A B (A ∩ B) ⊆ AX ⊆ Y ≡ ∀x. x ∈ X → x ∈ Y
A B x A ∩ BA ∩ B x ∈ A x ∈ B
26
![Page 65: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/65.jpg)
A simple set proof
Prove that for any sets and , we have .Recall that
Proof:Let and be arbitrary sets and an arbitrary element of .Then, by definition of , we have that and .It follows that , as required.
A B (A ∩ B) ⊆ AX ⊆ Y ≡ ∀x. x ∈ X → x ∈ Y
A B x A ∩ BA ∩ B x ∈ A x ∈ B
x ∈ A ◻
26
![Page 66: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/66.jpg)
Set proofs can use Boolean algebra equivalences
Prove that for any sets and , we have .
Proof:
A B (A ∩ B) ∪ (A ∩ ) = AB⎯ ⎯⎯⎯
27
![Page 67: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/67.jpg)
Set proofs can use Boolean algebra equivalences
Prove that for any sets and , we have .
Proof:Let and be arbitrary sets.
A B (A ∩ B) ∪ (A ∩ ) = AB⎯ ⎯⎯⎯
A B
27
![Page 68: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/68.jpg)
Set proofs can use Boolean algebra equivalences
Prove that for any sets and , we have .
Proof:Let and be arbitrary sets.Since set operations are defined using logical connectives, theequivalences of Boolean algebra can be used directly, as follows:
A B (A ∩ B) ∪ (A ∩ ) = AB⎯ ⎯⎯⎯
A B
27
![Page 69: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/69.jpg)
Set proofs can use Boolean algebra equivalences
Prove that for any sets and , we have .
Proof:Let and be arbitrary sets.Since set operations are defined using logical connectives, theequivalences of Boolean algebra can be used directly, as follows:
DistributivityComplementarityIdentity
A B (A ∩ B) ∪ (A ∩ ) = AB⎯ ⎯⎯⎯
A B
(A ∩ B) ∪ (A ∩ )B⎯ ⎯⎯⎯ = A ∩ (B ∪ )B
⎯ ⎯⎯⎯
= A ∩ U= A
27
![Page 70: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/70.jpg)
Set proofs can use Boolean algebra equivalences
Prove that for any sets and , we have .
Proof:Let and be arbitrary sets.Since set operations are defined using logical connectives, theequivalences of Boolean algebra can be used directly, as follows:
DistributivityComplementarityIdentity
A B (A ∩ B) ∪ (A ∩ ) = AB⎯ ⎯⎯⎯
A B
(A ∩ B) ∪ (A ∩ )B⎯ ⎯⎯⎯ = A ∩ (B ∪ )B
⎯ ⎯⎯⎯
= A ∩ U= A
Universe corresponds to 1 and corresponds to 0.U ∅
27
![Page 71: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/71.jpg)
More setsPower set, Cartesian product, and Russell’s paradox.
28
![Page 72: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/72.jpg)
Power set
Power set of a set is the set of all subsets of .
ExamplesLet .
A A(A) = {B : B ⊆ A}
Days = {M, W, F}(Days) =(∅) =
29
![Page 73: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/73.jpg)
Power set
Power set of a set is the set of all subsets of .
ExamplesLet .
A A(A) = {B : B ⊆ A}
Days = {M, W, F}(Days) = {∅, {M}, {W}, {F}, {M, W}, {M, F}, {W, F}, {M, W, F}}(∅) =
29
![Page 74: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/74.jpg)
Power set
Power set of a set is the set of all subsets of .
ExamplesLet .
A A(A) = {B : B ⊆ A}
Days = {M, W, F}(Days) = {∅, {M}, {W}, {F}, {M, W}, {M, F}, {W, F}, {M, W, F}}(∅) = {∅} ≠∅
29
![Page 75: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/75.jpg)
Cartesian product
The Cartesian product of two sets is the set of all of their ordered pairs.
Examples is the real plane. is the set of all pairs of integers.
A × B = {(a, b) : a ∈ A ∧ b ∈ B}
ℝ × ℝℤ × ℤ
30
![Page 76: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/76.jpg)
Cartesian product
The Cartesian product of two sets is the set of all of their ordered pairs.
Examples is the real plane. is the set of all pairs of integers.
If ,then
A × B = {(a, b) : a ∈ A ∧ b ∈ B}
ℝ × ℝℤ × ℤ
A = {1, 2}, B = {a, b, c}A × B =
30
![Page 77: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/77.jpg)
Cartesian product
The Cartesian product of two sets is the set of all of their ordered pairs.
Examples is the real plane. is the set of all pairs of integers.
If ,then .
A × B = {(a, b) : a ∈ A ∧ b ∈ B}
ℝ × ℝℤ × ℤ
A = {1, 2}, B = {a, b, c}A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
30
![Page 78: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/78.jpg)
Cartesian product
The Cartesian product of two sets is the set of all of their ordered pairs.
Examples is the real plane. is the set of all pairs of integers.
If ,then .
A × B = {(a, b) : a ∈ A ∧ b ∈ B}
ℝ × ℝℤ × ℤ
A = {1, 2}, B = {a, b, c}A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
A × ∅ =
30
![Page 79: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/79.jpg)
Cartesian product
The Cartesian product of two sets is the set of all of their ordered pairs.
Examples is the real plane. is the set of all pairs of integers.
If ,then .
.
A × B = {(a, b) : a ∈ A ∧ b ∈ B}
ℝ × ℝℤ × ℤ
A = {1, 2}, B = {a, b, c}A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
A × ∅ = {(a, b) : a ∈ A ∧ b ∈ ∅} = {(a, b) : a ∈ A ∧ '} = ∅
30
![Page 80: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/80.jpg)
Russell’s paradox
Let be the set of all sets that don’t contain themselves.SS = {x : x ∉ x}
31
![Page 81: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/81.jpg)
Russell’s paradox
Let be the set of all sets that don’t contain themselves.
The definition of is contradictory, hence the paradox.
SS = {x : x ∉ x}
S
31
![Page 82: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/82.jpg)
Russell’s paradox
Let be the set of all sets that don’t contain themselves.
The definition of is contradictory, hence the paradox.Suppose that . Then, by definition of , , which is acontradiction.
SS = {x : x ∉ x}
SS ∈ S S S ∉ S
31
![Page 83: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/83.jpg)
Russell’s paradox
Let be the set of all sets that don’t contain themselves.
The definition of is contradictory, hence the paradox.Suppose that . Then, by definition of , , which is acontradiction.Suppose that . Then, by definition of , , which is acontradiction too.
SS = {x : x ∉ x}
SS ∈ S S S ∉ S
S ∉ S S S ∈ S
31
![Page 84: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/84.jpg)
Russell’s paradox
Let be the set of all sets that don’t contain themselves.
The definition of is contradictory, hence the paradox.Suppose that . Then, by definition of , , which is acontradiction.Suppose that . Then, by definition of , , which is acontradiction too.
To avoid the paradox …Define with respect to a universe of discourse.
With this definition, and there is no contradiction because .
SS = {x : x ∉ x}
SS ∈ S S S ∉ S
S ∉ S S S ∈ S
SS = {x ∈ U : x ∉ x}
S ∉ S S ∉ U
31
![Page 85: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/85.jpg)
Working with setsRepresenting sets as bitvectors and applications of bitvectors.
32
![Page 86: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/86.jpg)
Representing sets as bitvectorsSuppose that universe is .
We can represent every set as a vector of bits:
This is called the characteristic vector of set .
U {1, 2, … , n}B ⊆ U
… where b1b2 bn = 1 if i ∈ Bbi= 0 if i ∉ Bbi
B
33
![Page 87: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/87.jpg)
Representing sets as bitvectorsSuppose that universe is .
We can represent every set as a vector of bits:
This is called the characteristic vector of set .
Given characteristic vectors for and , what is the vector for
U {1, 2, … , n}B ⊆ U
… where b1b2 bn = 1 if i ∈ Bbi= 0 if i ∉ Bbi
B
A BA ∪ B =A ∩ B =
33
![Page 88: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/88.jpg)
Representing sets as bitvectorsSuppose that universe is .
We can represent every set as a vector of bits:
This is called the characteristic vector of set .
Given characteristic vectors for and , what is the vector for
U {1, 2, … , n}B ⊆ U
… where b1b2 bn = 1 if i ∈ Bbi= 0 if i ∉ Bbi
B
A BA ∪ B = ( ∨ ) … ( ∨ )a1 b1 an bnA ∩ B =
33
![Page 89: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/89.jpg)
Representing sets as bitvectorsSuppose that universe is .
We can represent every set as a vector of bits:
This is called the characteristic vector of set .
Given characteristic vectors for and , what is the vector for
U {1, 2, … , n}B ⊆ U
… where b1b2 bn = 1 if i ∈ Bbi= 0 if i ∉ Bbi
B
A BA ∪ B = ( ∨ ) … ( ∨ )a1 b1 an bnA ∩ B = ( ∧ ) … ( ∧ )a1 b1 an bn
33
![Page 90: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/90.jpg)
Unix/Linux file permissions$ ls -ldrwxr-xr-x ... Documents/-rw-r--r-- ... file1
Permissions maintained as bitvectors.Letter means the bit is 1.”-“ means the bit is zero.
34
![Page 91: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/91.jpg)
Bitwise operations
Note that .
01101101∨ 00110111
01111111
z = x | y
00101010∧ 00001111
00001010
z = x & y
01101101⊕ 00110111
01011010
z = x ^ y (x ⊕ y) ⊕ y = x
35
![Page 92: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/92.jpg)
Private key cryptographyAlice wants to communicate a message secretly to Bob, so thateavesdropper Eve who sees their conversation can’t understand .
Alice and Bob can get together ahead of time and privately share asecret key .
How can they communicate securely in this setting?
Sender(Alice)
plaintextmessage key
encryptReceiver(Bob)
plaintextmessagekey
decryptcyphertext
Eavesdropper(Eve)
mm
K
36
![Page 93: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/93.jpg)
One-time pad
Alice and Bob privately share a random -bitvector .Eve doesn’t know .
Later, Alice has -bit message to send to Bob.Alice computes .Alice sends to Bob.Bob computes , which is .
Eve can’t figure out from unless she can guess .And that’s very unlikely for large …
n KK
n mC = m ⊕ K
Cm = C ⊕ K (m ⊕ K) ⊕ K = m
m C Kn
37
![Page 94: CSE 311 Lecture 10: Set Theory - University of WashingtonFun strategy: proof by computer Use an automated theorem prover:; No integer is both even and odd. ... A set is a collection](https://reader033.vdocuments.mx/reader033/viewer/2022042401/5f103fe37e708231d4482d99/html5/thumbnails/94.jpg)
SummarySets are a basic notion in mathematics and computer science.
Collections of objects called elements.Can be compared for equality ( ) and containment ( ).
Set operations correspond to Boolean algebra operations.You can prove theorems about sets using Boolean algebra laws.
Sets can be represented efficiently using bitvectors.This representation is used heavily in the real world.With this representation, set operations reduce to fast bitwise operations.
= ⊆
38