Discrete Math

Question 1

Assume propositions p, q, and r have the following truth values:

p is true
q is true
r is false

2)
¬r
True
False

Answer 1

true
Correct
The negation of a false proposition is true.

Question 2

p is true
q is true
r is false

3)
p ∧ r
True
False

Answer 2

false
Correct
Since r is false, the proposition is false.

Question 3

Assume the propositions p, q, r, and s have the following truth values:

p is false
q is true
r is false
s is true

What are the truth values for the following compound propositions?
(b)

p ∨ r

Answer 3

False

p ∨ q is false only when p and q are both false.

Question 4

Assume the propositions p, q, r, and s have the following truth values:

p is false
q is true
r is false
s is true

What are the truth values for the following compound propositions?

q ⊕ r

Answer 4

True

The exclusive or operation is usually denoted with the symbol ⊕. The proposition p ⊕ q is true if exactly one of the propositions p or q is true but not both. This question asks you to fill in the truth table for p ⊕ q.

Question 5

Indicate whether each statement is true or false, assuming that the “or” in the sentence means the inclusive or. Then indicate whether the statement is true or false if the “or” means the exclusive or.

Answer 5

(a)

February has 31 days or the number 5 is an integer.

Inclusive or: True. Exclusive or: True.

Note: evaluates to true when p is true and q is false or when q is true and p is false.
- only when one of the propositions is true, but not both.

Note: evaluates to true when one or both of the propositions are true.
- when at least one of the propositions is true.

Question 6

In this equation for logic math : TRUE ∧ ¬(FALSE ∧ TRUE), after following order of operations how to FALSE cancel out to equal the next step T ∧ ¬ T / F . Did it cancel out?

Answer 6

Let’s break it down step by step using logical operations:
- Start with the given expression:
[ TRUE \land \neg (FALSE \land TRUE) ]
- Evaluate the inner parentheses:
- FALSE ∧ TRUE evaluates to FALSE (since AND requires both to be true).
- Now the expression simplifies to:
[ TRUE \land \neg FALSE ]
- Apply negation (¬):
- ¬FALSE is TRUE (since NOT flips the truth value).
- Now the expression simplifies to:
[ TRUE \land TRUE ]
- Evaluate the final AND operation:
- TRUE ∧ TRUE evaluates to TRUE.
So, the FALSE didn’t “cancel out”—it was negated and transformed into TRUE. The final result is TRUE

Question 7

This question uses the following propositions:

p: I will share my cookie with you.
q: You will share your soda with me.

Select the conditional statement that has the same logical meaning as the English sentence given.

I will share my cookie with you only if you share your soda with me.
q → p
p → q

Answer 7

p only if q.
Correct