Question

The propositional variables, p, q, and s have the following truth assignments: p = T, q = T, s = F. Give the truth value for each proposition. (a) p ∧ ¬(q ∨ s) (b) ¬(q ∧ p ∧ ¬s)

Answer 1

Answer: a) FALSE b) FALSE

Explanation:

a) For the given proposition p ∧ ¬(q ∨ s) you can solve first (q v s)

q v s is true if either q is true or s is true or both. It is only false if both q and s are false. So, the proposition (q v s) is true because q is true.

Now you can solve the negation: ¬(q ∨ s)

As we know, (q v s) is true then its negation ¬(q ∨ s) is false.

p ∧ ¬(q ∨ s) should be true when both p and ¬(q ∨ s) are true, and false otherwise. So, the proposition is false because p is true and ¬(q ∨ s) is false.

b) For the given proposition ¬(q ∧ p ∧ ¬s)

You can rewrite the expression as: ¬[q ∧ (p ∧ ¬s) ] to solve first each part of the propositions in parenthesis.

The negation of s: ¬s is true because s is false

Now, you can solve (p ∧ ¬s) which is true because both p and ¬s are true.

To continue, you have to solve (q ∧ p ∧ ¬s) which is true because both q and (p ∧ ¬s) are true.

To finish, the negation: ¬(q ∧ p ∧ ¬s) is false.