Deck 1: The Foundations: Logic and Proofs

Question 1

What is the negation of the propositions in
Abby has more than 300 friends on facebook.

Answer 1

Abby has fewer than 301 friends on facebook.

Question 2

(a) Find a proposition with the truth table at the right.
(b) Find a proposition using only and the connective that has this truth table.

Answer 2

(a)
(b)

Question 3

In questions , determine whether the proposition is TRUE or FALSE.
1 + 1 = 3 if and only if 2 + 2 = 3.

Answer 3

True

Question 4

In questions , determine whether the proposition is TRUE or FALSE.
If it is raining, then it is raining.

Question 5

Find a proposition with three variables p, q, and r that is never true.

Question 6

Write a proposition equivalent to that uses only and the connective

Question 7

Find a proposition with three variables p, q, and r that is true when at most one of the three variables is true,
and false otherwise.

Question 8

What is the negation of the propositions in
A messaging package for a cell phone costs less than $20 per month.

Question 9

In questions , determine whether the proposition is TRUE or FALSE.
If 2 + 1 = 3, then 2 = 3 − 1.

Question 10

Determine whether

Question 11

Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and false
otherwise.

Question 12

Determine whether

Question 13

Write a proposition equivalent to and the connective

Question 14

Determine whether

Question 15

In questions , determine whether the proposition is TRUE or FALSE.
If 1 < 0, then 3 = 4.

Question 16

In questions , determine whether the proposition is TRUE or FALSE.
If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4.

Question 17

Write the truth table for the proposition

Question 18

What is the negation of the propositions in
4.5 + 2.5 = 6

Question 19

Question 20

Prove that the proposition "if it is not hot, then it is hot" is equivalent to "it is hot".

Question 21

Write a proposition equivalent to using only , and the connective .

Question 22

write the statement in the form "If . . . , then . . . ."
The team wins if the quarterback can pass.

Question 23

write the statement in the form "If . . . , then . . . ."
It is hot whenever it is sunny.

Question 24

Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday.

Question 25

Determine whether the following two propositions are logically equivalent:

Question 26

Write a proposition equivalent to using only , and the connective .

Question 27

Prove that and its inverse are not logically equivalent.

Question 28

write the negation of the statement. (Don't write "It is not true that . . . .")
It is Thursday and it is cold.

Question 29

Prove that is a tautology using propositional equivalence and the laws of logic.

Question 30

write the statement in the form "If . . . , then . . . ."
You need to be registered in order to check out library books.

Question 31

Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win.

Question 32

write the statement in the form "If . . . , then . . . ."
To get a good grade it is necessary that you study.

Question 33

write the statement in the form "If . . . , then . . . ."
Studying is sufficient for passing.

Question 34

write the negation of the statement. (Don't write "It is not true that . . . .")
I will go to the play or read a book, but not both.

Question 35

Determine whether this proposition is a tautology:

Question 36

Prove that and its converse are not logically equivalent.

Question 37

write the statement in the form "If . . . , then . . . ."

Question 38

write the statement in the form "If . . . , then . . . ."

Question 39

Determine whether the following two propositions are logically equivalent:

Question 40

Determine whether this proposition is a tautology:

Question 41

Using c for "it is cold", r for "it is rainy", and w for "it is windy", write "It is rainy only if it is windy and
cold" in symbols.

Question 42

Question 43

On the island of knights and knaves you encounter two people, A and
B. Person A says "B is a knave."
Person B says "At least one of us is a knight." Determine whether each person is a knight or a knave.
Exercises 51-53 relate to inhabitants of an island on which there are three kinds of people: knights who always
tell the truth, knaves who always lie, and spies who can either tell the truth or lie. You encounter three people,
A, B, and
C. You know one of the three people is a knight, one is a knave, and one is a spy. Each of the three

Question 44

On the island of knights and knaves you encounter two people, A and
B. Person A says "B is a knave."
Person B says "We are both knights." Determine whether each person is a knight or a knave.

Question 45

Explain why the negation of "Al and Bill are absent" is not "Al and Bill are present".

Question 46

people knows the type of person each of the other two is. For each of these situations, if possible, determine
whether there is a unique solution, list all possible solutions or state that there are no solutions.

Question 47

people knows the type of person each of the other two is. For each of these situations, if possible, determine
whether there is a unique solution, list all possible solutions or state that there are no solutions.

Question 48

Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs in from input bits p, q and r.

Question 49

Using c for "it is cold" and r for "it is rainy", write "It is rainy if it is not cold" in symbols.

Question 50

people knows the type of person each of the other two is. For each of these situations, if possible, determine
whether there is a unique solution, list all possible solutions or state that there are no solutions.

Question 51

Using c for "it is cold" and w for "it is windy", write "To be windy it is necessary that it be cold" in symbols.

Question 52

Question 53

Translate the given statement into propositional logic using the propositions provided: On certain highways
in the Washington, DC metro area you are allowed to travel on high occupancy lanes during rush hour only
if there are at least three passengers in the vehicle. Express your answer in terms of r:"You are traveling
during rush hour." t:"You are riding in a car with at least three passengers." and h:"You can travel on a high
occupancy lane."

Question 54

A set of propositions is consistent if there is an assignment of truth values to each of the variables in the
propositions that makes each proposition true. Is the following set of propositions consistent?
The system is in multiuser state if and only if it is operating normally.
If the system is operating normally, the kernel is functioning.
The kernel is not functioning or the system is in interrupt mode.
If the system is not in multiuser state, then it is in interrupt mode.
The system is in interrupt mode.

Question 55

suppose that where x is a real number. Find the truth value of the statement.

Question 56

Question 57

Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs in from input bits p, q and r.
Determine whether the compound propositions in 58-59 are satisfiable.

Question 58

write the negation of the statement. (Don't write "It is not true that . . . .")
If it is rainy, then we go to the movies.

Question 59

Find the output of the combinatorial circuits in 54-55.

Question 60

Using c for "it is cold" and d for "it is dry", write "It is neither cold nor dry" in symbols.

Question 61

P(x, y) means "x and y are real numbers such that Determine whether the statement
is true.
In 73-75 P(m, n) means where the universe of discourse for m and n is the set of nonnegative
integers. What is the truth value of the statement?

Question 62

P(m, n) means

where the universe of discourse for m and n is the set of nonnegative
integers. What is the truth value of the statement?

Question 63

suppose that where x is a real number. Find the truth value of the statement.

Question 64

P(x, y) means
where x and y are integers. Determine the truth value of the statement.
P(0, 0).

Question 65

P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether
the following statements are true.

Question 66

P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether
the following statements are true.

Question 67

P(x, y) means
where x and y are integers. Determine the truth value of the statement.

Question 68

P(m, n) means

where the universe of discourse for m and n is the set of nonnegative
integers. What is the truth value of the statement?

Question 69

P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether
the following statements are true.

Question 70

P(x, y) means
where x and y are integers. Determine the truth value of the statement.

Question 71

suppose that where x is a real number. Find the truth value of the statement.
In 63-70 P(x, y) means where x and y are integers. Determine the truth value of the statement.

Question 72

P(x, y) means
where x and y are integers. Determine the truth value of the statement.

Question 73

P(x, y) means
where x and y are integers. Determine the truth value of the statement.

Question 74

P(x, y) means
where x and y are integers. Determine the truth value of the statement.

Question 75

P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether
the following statements are true.

Question 76

P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose
P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether
the following statements are true.

Question 77

P(x, y) means "x and y are real numbers such that Determine whether the statement
is true.

Question 78

P(m, n) means

where the universe of discourse for m and n is the set of nonnegative
integers. What is the truth value of the statement?

Question 79

P(x, y) means
where x and y are integers. Determine the truth value of the statement.

Question 80

P(x, y) means
where x and y are integers. Determine the truth value of the statement.