Set Operations
Niloufar Shafiei
1
Set operations
Two sets can be combined in many
different ways.
Set operations can be used to combine
sets.
2
Union
Let A and B be sets.
The union of A and B, denoted by A B, is
the set containing those elements that are
either in A or in B, or in both.
A B = {x | x A x B}
UB A
3
Union (example)
{1,2,3} {2,4,6,8} =
{1,2,3,4,6,8}
{x| x Z x is even} {x|x Z x is odd} =
Z
4
Intersection
Let A and B be sets.
The intersection of A and B, denoted by A
B, is the set containing those elements in
both A and B.
A B = {x | x A x B}
UB A
5
Intersection (example)
{1,2,3} {2,4,6,8} =
{2}
Z {x|x Z x is odd} =
{x|x Z x is odd}
{x|x Z x is even} {x|x Z x is odd} =
Ø
Two sets are called disjoint if their intersection is empty.
6
The cardinality of the union of sets
|A B|=?
Solution:
A={1,2,3} B={2,3,4} A B={1,2,3,4}
|A|=3 |B|=3 A B|=4
|A B| = |A| + |B| - |A B|
Principle of inclusion-exclusion
7
Difference
Let A and B be sets.
The difference of A and B, denoted by A-B,
is the set containing those elements that
are in A but not in B. (also called complement
of B with respect to A)
A-B = {x | x A x B}
UA B
8
Difference (example)
{1,2,3} - {2,4} =
{1,3}
Z - {x|x Z x is odd} =
{x|x Z x is even}
9
Complement
Let U be the universal set and A be a set.
The complement of A, denoted by , is the
complement of A with respect to U (which
is U-A).
= {x | x A}
UA
10
Complement (example)
A={a,b,c,d} and U is the set of English
alphabet
= {e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
A = {x|x Z x is odd} and U is Z
= {x|x Z x is even}
11
Set identities
A = A
A U = A
A U = U
A =
A A = A
A A = A
( ) = A
A = U
A =
UA
Identity laws
Domination laws
Idempotent laws
Complementation law
Complement laws
12
Set identities
A B = B A
A B = B A
Commutative laws
A (B C) = (A B) C
A (B C) = (A B) C
Associative laws
A (A B) = A
A (A B) = A
Absorption laws
UA B C
13
Set identities (example)
Show A B = A B.
Solution:
A B = {x | x A B }
= {x | ¬(x A B) }
= {x | ¬((x A) (x B)) }
= {x | ¬(x A) ¬(x B) }
= {x | (x A) (x B) }
= {x | (x A) (x B) }
= {x | (x A B)}
= A B
14
Set identities
A B = A B
A B = A B
De Morgan’s laws
There is the similarity between set identities andlogical equivalences.
¬(A B) ¬A ¬B
¬(A B) ¬A ¬B
15
Set identities (example)
Show A (B C) = (A B) (A C).
Solution:
Part 1: A (B C) (A B) (A C)
Assume x A (B C).
(x A) (x (B C))
(x A) (x B r x C)
(x A x B) (x A x C)
(x (A B)) (x (A C))
x (A B) (A C)
So, A (B C) (A B) (A C).
16
Set identities (example)
Show A (B C) = (A B) (A C).
Solution:
Part 2: (A B) (A C) A (B C)
Assume x (A B) (A C).
x (A B) r x (A C)
(x A x B) (x A x C)
x A (x B x C)
x A (x (B C))
x A (B C)
So, (A B) (A C) A (B C) .
Thus, A (B C) = (A B) (A C).
17
Set identities
A (B C) = (A B) (A C)
A (B C) = (A B) (A C)
Distributive laws
There is the similarity between set identities
and logical equivalences.
A (B C) (A B) (A C)
A (B C) (A B) (A C)
18
Set identities
A (B C) = (A B) (A C)
A (B C) = (A B) (A C)
Distributive laws
000000 0 0
010000 0 1
001000 1 0
111110 1 1
111101 0 0
111101 0 1
111101 1 0
111111 1 1
(A B) (A C)A CA BA (B C)B CA B C
Membership table
19
Set identities (example)
Show A (B C) = (C B) A.
Solution:
A (B C) =
A (B C) = (by De Morgan’s law)
A (B C) = (by De Morgan’s law)
(C B) A (by commutative law)
20
Database query
The query that returns students that their GPA
is more than B and they are either
computer science or mathematics major.
A: students that their GPA is more than B
B: students that are computer science major.
C: students that are Mathematics major.
A (B C)
21
Generalized union
Assume A1, A2, … and An are sets
The union of A1, A2, … and An is the set that
contains those elements that are members
of at least one set.
A1 A2 … An = Ai
n
i=1
22
Generalized union (example)
Assume Ai is {i, i+1, i+2, …}. What is Ai?
Solution:
A1 = {1,2,3,…}
A2 = {2,3,4,…}
A3 = {3,4,5,…}
: Ai = {1,2,3,…}
n
i=1
n
i=1
23
Generalized intersection
Assume A1, A2, … and An are sets
The intersection of A1, A2, … and An is theset that contains those elements that aremembers of all sets.
A1 A2 … An = Ai
n
i=1
24
Generalized intersection (example)
Assume Ai is {i, i+1, i+2, …}. What is Ai?
Solution:
A1 = {1,2,3,…}
A2 = {2,3,4,…}
:
An = {n,n+1,n+2,…}
Ai = {n,n+1,n+2,…}
n
i=1
n
i=1
25
Generalized union and intersection
A1 A2 … An … = Ai
A1 A2 … An … = Ai
i=1
i=1
26
Example
Assume Ai = {1,2,3,…,i}. What is Ai and Ai?
Solution:
A1 = {1}
A2 = {1,2}
A3 = {1,2,3}
: Ai = Z+
Ai = {1}
i=1i=1
i=1
i=1
27
Recommended exercises
3,5,7,9,11,13,17,19,24,27,29,45,49