Unit 1- LogicExercises

2• Which of these sentences are propositions? What are the truth values

of those that are propositions?a) Boston is the capital of Massachusetts.b) Miami is the capital of Florida.c) 2 + 3 = 5.d) 5 + 7 = 10.e) x + 2 = 11.f ) Answer this question

3

• Which of these are propositions? What are the truth values of those that are propositions?

a) Do not pass go.b) What time is it?c) There are no black flies in Mained) 4 + x = 5.e) The moon is made of green cheese.f ) 2n ≥ 100.

4

• What is the negation of each of these propositions?a) Mei has an MP3 player.b) There is no pollution in New Jersey.c) 2 + 1 = 3.d) The summer in Maine is hot and sunny.

5

• What is the negation of each of these propositions?a) Jennifer and Teja are friends.b) There are 13 items in a baker’s dozen.c) Abby sent more than 100 text messages every day.d) 121 is a perfect square.

6

• What is the negation of each of these propositions?a) Steve has more than 100 GB free disk space on hislaptop.b) Zach blocks e-mails and texts from Jennifer.c) 7 · 11 · 13 = 999.d) Diane rode her bicycle 100 miles on Sunday.

7• Suppose that Smartphone A has 256MB RAM and 32GB ROM, and the

resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP; and Smartphone C has 128 MB RAM and 32 GB ROM, and the resolution of its camera is 5 MP. Determine the truth value of each of these propositions.

• a) Smartphone B has the most RAM of these three smartphones.• b) Smartphone C has more ROM or a higher resolution camera than

Smartphone B.• c) Smartphone B has more RAM, more ROM, and a higher resolution

camera than Smartphone A.• d) If Smartphone B has more RAM and more ROM than Smartphone C,

then it also has a higher resolution camera.• e) Smartphone A has more RAM than Smartphone B if and only if

Smartphone B has more RAM than Smartphone A

8

• Let p and q be the propositions• p : I bought a lottery ticket this week.• q : I won the million dollar jackpot.• Express each of these propositions as an English sentence.• a) ￢ p b) p ∨ q c) p → q• d) p ∧ q e) p ↔ q f ) ￢ p →￢ q• g) ￢ q ∧￢ p h) ￢ p ∨ (p ∧ q)

9

• Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence.

a) ￢ q b) p ∧ q c) ￢ p ∨ qd) p →￢ q e) ￢ q → p f ) ￢ p →￢ qg) p ↔￢ q h) ￢ p ∧ (p∨ ￢ q)

10

• Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express each of these compound propositions as an English sentence.

a) ￢ p b) p ∨ qc) ￢ p ∧ q d) q → pe) ￢ q →￢ p f ) ￢ p →￢ qg) p ↔ q h) ￢ q ∨ (￢ p ∧ q)

11

• Let p, q, and r be the propositions• p :You have the flu.• q :You miss the final examination.• r :You pass the course.• Express each of these propositions as an English sentence.a) p → q b) ￢ q ↔ rc) q →￢ r d) p ∨ q ∨ re) (p →￢ r) ∨ (q →￢ r)f ) (p ∧ q) ∨ (￢ q ∧ r)

12

• Construct a truth table for each of these compound propositions.

a) p ∧￢ p b) p ∨￢ pc) (p ∨￢ q) → q d) (p ∨ q) → (p ∧ q)e) (p → q) ↔ (￢ q →￢ p)f ) (p → q) → (q → p)

13

• Construct a truth table for each of these compound propositions.• a) p →￢ p b) p ↔￢ p• c) p ⊕ (p ∨ q) d) (p ∧ q) → (p ∨ q)• e) (q →￢ p) ↔ (p ↔ q)• f ) (p ↔ q) ⊕ (p ↔￢ q)

14

• Construct a truth table for each of these compound propositions.• a) (p ∨ q) → (p ⊕ q) b) (p ⊕ q) → (p ∧ q)• c) (p ∨ q) ⊕ (p ∧ q) d) (p ↔ q) ⊕ (￢ p ↔ q)• e) (p ↔ q) ⊕ (￢ p ↔￢ r)• f ) (p ⊕ q) → (p ⊕￢ q)

15

• Construct a truth table for each of these compound propositions.• a) p ⊕ p b) p ⊕￢ p• c) p ⊕￢ q d) ￢ p ⊕￢ q• e) (p ⊕ q) ∨ (p ⊕￢ q) f ) (p ⊕ q) ∧ (p ⊕￢ q)

16

• Construct a truth table for each of these compound propositions.• a) p →￢ q b) ￢ p ↔ q• c) (p → q) ∨ (￢ p → q) d) (p → q) ∧ (￢ p → q)• e) (p ↔ q) ∨ (￢ p ↔ q)• f ) (￢ p ↔￢ q) ↔ (p ↔ q)

17

• Construct a truth table for each of these compound propositions.• a) (p ∨ q) ∨ r b) (p ∨ q) ∧ r• c) (p ∧ q) ∨ r d) (p ∧ q) ∧ r• e) (p ∨ q)∧￢ r f ) (p ∧ q)∨￢ r

18

• Construct a truth table for each of these compound propositions.• a) p → (￢ q ∨ r)• b) ￢ p → (q → r)• c) (p → q) ∨ (￢ p → r)• d) (p → q) ∧ (￢ p → r)• e) (p ↔ q) ∨ (￢ q ↔ r)• f ) (￢ p ↔￢ q) ↔ (q ↔ r)

19Construct a truth table for

•((p → q) → r) → s•(p ↔ q) ↔ (r ↔ s)

20

• Use De-Morgans law to express the negations of “Miguel has a cell phone and he has a laptop computer” and “Heather will go to concert or Steve will go to concert”

21

• Show that ￢ (p → q) and p ∧￢ q are logically equivalent using laws• Show that ￢ (p ∨ (￢ p ∧ q)) and ￢ p ∧￢ q are logically

equivalent by developing a series of logical equivalences.• Show that (p ∧ q) → (p ∨ q) is a tautology.

22

• Show that each of these conditional statements is a tautology by using truth tables.• a) (p ∧ q) → p b) p → (p ∨ q)• c) ￢ p → (p → q d) (p ∧ q) → (p → q)• e) ￢ (p → q) → p f ) ￢ (p → q)→￢ q

23

• Show that each of these conditional statements is a tautology by using truth tables.• a) [￢ p ∧ (p ∨ q)] → q• b) [(p → q) ∧ (q → r)] → (p → r)• c) [p ∧ (p → q)] → q• d) [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r

24Subject - predicates

• x is greater than 3” has two parts. • The first part, the variable x, is the subject of the statement. • The second part—the predicate, “is greater than 3”—refers

to a property that the subject of the statement can have.• We can denote the statement “x is greater than 3” by P(x),

where P denotes the predicate “is greater than 3” and x is the variable. • The statement P(x) is also said to be the value of the

propositional function P at x

25

• Let P(x) denote the statement “x > 3.” What are the truth values of P(4) and P(2)?

26Try this

1. Let P(x) denote the statement “x ≤ 4.” What are these truth values?

a) P(0) b) P(4) c) P(6)2. Let P(x) be the statement “the word x contains the letter a.” What are these truth values?a) P(orange) b) P(lemon)c) P(true) d) P(false)

27

• Let R(x, y, z) denote the statement`‘ x + y = z.” When values are assigned to the variables x, y, and z, this statement has a truth value. What are the truth values of the propositions R(1, 2, 3) and R(0, 0, 1)?

28

• Let A(x) denote the statement “Computer x is under attack by an intruder.” Suppose that of the computers on campus, only CS2 and MATH1 are currently under attack by intruders. What are truth values of A(CS1), A(CS2), and A(MATH1)?

29

• Let Q(x, y) denote the statement “x = y + 3.” What are the truth values of the propositions Q(1, 2) and Q(3, 0)?

30

• if x > 0 then x := x + 1

31

• Let A(c, n) denote the statement “Computer c is connected to network n,” where c is a variable representing a computer and n is a variable representing a network. Suppose that the computer MATH1 is connected to network CAMPUS2, but not to network CAMPUS1. What are the values of A(MATH1, CAMPUS1) and A(MATH1, CAMPUS2)?

32Try this one

State the value of x after the statement ifP(x) then x := 1 is executed, where P(x) is the statement “x > 1,” if the value of x when this statement is reached is• a) x = 0 b) x = 1• c) x = 2

33Preconditions and postcondition

• The statements that describe valid input are known as preconditions • The conditions that the output should satisfy when the

program has run are known as postconditions.

34

• Consider the following program, designed to interchange the values of two variables x and y.

temp := xx := y

y := temp• Find predicates that we can use as the precondition and the

postcondition to verify the correctness of this program.

35Quantifiers• Quantification expresses the extent to which a

predicate is true over a range of elements.• Two types of quantification• Universal quantification: which tells us that a

predicate is true for every element under consideration• Existential quantification, which tells us that there is

one or more element under consideration for which the predicate is true.

• The area of logic that deals with predicates and quantifiers is called the predicate calculus

36

• The universal quantification of P(x) is the statement• “P(x) for all values of x in the domain.”

• The notation ∀ x P(x) denotes the universal quantification of P(x). Here ∀ is called the universal quantifier.• We read ∀xP(x) as “ for all x P(x)” or “for every xP(x).”• An element for which P(x) is false is called a

counterexample of ∀xP(x).

37

38example

• Let P(x) be the statement “x + 1 > x.” What is the truth value of the quantification ∀xP(x), where the domain consists of all real numbers?

• Let Q(x) be the statement “x < 2.” What is the truth value of the quantification ∀xQ(x), where the domain consists of all real numbers?

39

• When all the elements in the domain can be listed—say, x1, x2, . . ., xn—it follows that the universal quantification ∀xP(x) is the same as the conjunction

P(x1) ∧ P(x2) ∧ · · · ∧ P(xn),because this conjunction is true if and only if P(x1), P(x2), . . . , P (xn) are all true.

40

• What is the truth value of ∀x P(x), where P(x) is the statement “x2 < 10” and the domain consists of the positive integers not exceeding 4?

• What does the statement ∀x N(x) mean if N(x) is “Computer x is connected to the network” and the domain consists of all computers on campus?

41Existential quantification

• There exists an element x in the domain such that p(x)• There is an x such that P(x)• There is at least one x such that P(x)• the statement ∃xP(x) is false if and only if there is no

element x in the domain for which P(x) is true. • That is, ∃xP(x) is false if and only if P(x) is false for every

element of the domain

42examples

• Let P(x) denote the statement “x > 3.” What is the truth value of the quantification ∃xP(x), where the domain consists of all real numbers?

43

• Let Q(x) denote the statement “x = x + 1.”What is the truth value of the quantification ∃x Q(x), where the domain consists of all real numbers?

44

• When all elements in the domain can be listed—say, x1, x2, . . . , xn—the existential quantification ∃xP(x) is the same as the disjunction

P(x1) ∨ P(x2) ∨ · · · ∨ P(xn),because this disjunction is true if and only if at least one of P(x1), P(x2), . . . , P (xn) is true.

45

• What is the truth value of ∃x P(x), where P(x) is the statement “x2 > 10” and the universe of discourse consists of the positive integers not exceeding 4?