Deck 1: A: the Foundations: Logic and Proofs

Question 1

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

Answer 1

Question 2

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

Answer 2

True

Question 3

Write the truth table for the proposition

Answer 3

Question 4

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

Question 5

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 6

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

Question 7

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

Question 8

use the conditional-disjunction equivalence to find an equivalent compound proposition that does not
involve conditions.

Question 9

use the conditional-disjunction equivalence to find an equivalent compound proposition that does not
involve conditions.

Question 10

Determine whether and are equivalent.

Question 11

(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.

Question 12

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

Question 13

Find a proposition using only , and the connective with the truth table at the right.

Question 14

Determine whether .

Question 15

What is the negation of the propositions
Alissa owns more quilts than Federico.

Question 16

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

Question 17

Determine whether is equivalent to

Question 18

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

Question 19

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

Question 20

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

Question 21

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

Question 22

Question 23

Write a proposition equivalent to that uses only , and the connective

Question 24

Determine whether this proposition is a tautology:

Question 25

Prove that ¬p → ¬q and its inverse are not logically equivalent.

Question 26

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

Question 27

Determine whether this proposition is a tautology:

Question 28

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

Question 29

Write a proposition equivalent to using only , and the connectivehttps://d2lvgg3v3hfg70.cloudfront.net/TB34225555/.

Question 30

write the statement in the form "If . . . , then . . . ."
x is even only if y is odd.

Question 31

write the statement in the form "If . . . , then . . . ."
A implies B.

Question 32

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

Question 33

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

Question 34

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

Question 35

Determine whether the following two propositions are logically equivalent: https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/.

Question 36

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

Question 37

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

Question 38

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

Question 39

Write a proposition equivalent to using only , and the connective https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/.

Question 40

Prove that p → q and its converse are not logically equivalent.

Question 41

Express r ⊕ d in English, where r is "it is rainy" and d is "it is dry."

Question 42

A says "I am not a knight," B says "I am not a spy," and C says "I am not a knave."

Question 43

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 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 "At least one of us is a knight." Determine whether each person is a knight or a knave. Questions 56-58 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 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 45

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

Question 46

Find the output of the combinatorial circuits

Question 47

A says "I am a spy," B says "I am a spy" and C says "B is a spy."

Question 48

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 49

Find the output of the combinatorial circuits

Question 50

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

Question 51

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 52

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 53

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 54

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

Question 55

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 56

A says "I am a knight," B says "I am a knave," and C says "I am not a knave."

Question 57

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

Question 58

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 59

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

Question 60

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

Question 61

suppose that Q(x) is "x + 1 = 2x," where x is a real number. Find the truth value of the statement.

Question 62

suppose 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 63

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.
P(1, −1)

Question 64

P(m, n) means "m ≤ n," where the universe of discourse for m and n is the set of nonnegative integers.
What is the truth value of the statement?

Question 65

P(x, y) means "x and y are real numbers such that x + 2y = 5." Determine whether the statement is true.

Question 66

P(m, n) means "m ≤ n," where the universe of discourse for m and n is the set of nonnegative integers.
What is the truth value of the statement?

Question 67

express the negation of the statement in terms of quantifiers without using the negation symbol.

Question 68

suppose that Q(x) is "x + 1 = 2x," where x is a real number. Find the truth value of the statement.

Question 69

suppose that Q(x) is "x + 1 = 2x," where x is a real number. Find the truth value of the statement.

Question 70

express the negation of the statement in terms of quantifiers without using the negation symbol.

Question 71

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.
P(0, 0)

Question 72

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.

Question 73

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.

Question 74

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.

Question 75

P(x, y) means "x and y are real numbers such that x + 2y = 5." Determine whether the statement is true.

Question 76

P(m, n) means "m ≤ n," where the universe of discourse for m and n is the set of nonnegative integers.
What is the truth value of the statement?

Question 77

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.

Question 78

Determine whether the compound propositions are satisfiable.

Question 79

Determine whether the compound propositions are satisfiable.

Question 80

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.