Question

use the statements to write a compound statement for the conjunction or disjunction. then find the truth values.
p: - 3 - 2 = - 5
q: vertical angles are congruent.
r: 2 + 8 > 10
~ p ∧ ~ q

• 3 - 2 ≠ - 5, and vertical angles are congruent; ~ p is false, and ~ q is false, so ~ p ∧ ~ q is false.

Explanation:

Step1: Determine the truth value of \(p\)

Given \(p: - 3-2=-5\). Calculate \(-3 - 2=-5\), so \(p\) is true. Then \(\sim p\) (the negation of \(p\)) is false.

Step2: Determine the truth value of \(q\)

Given \(q\): "Vertical angles are congruent". By the vertical - angle theorem, \(q\) is true. Then \(\sim q\) (the negation of \(q\)) is false.

Step3: Analyze the compound statement \(\sim p\wedge\sim q\)

The logical "and" (\(\wedge\)) operation: For \(a\wedge b\), the compound statement is true if and only if both \(a\) and \(b\) are true. Here \(a = \sim p\) (false) and \(b=\sim q\) (false). So \(\sim p\wedge\sim q\) is false.

Answer:

\(\sim p\) is false, \(\sim q\) is false, and \(\sim p\wedge\sim q\) is false.