5.3 Exercise Let P(z) be the predicate "r is a prime number" and the domain of discourse be the set of positive integers Determine the truth value of the following statement: Vr P(x). 5.4 Exercise Let Q(z) be the predicate " is a perfect square" and the domain of discourse be the set of integers. Determin the truth value of the following statement: Er Q(z). 5.5 Exercise Let R(x, y) be the predicate "z divides y" and the domain of discourse be the set of positive integers. Determin the truth value of the following statement: Vray R(x,y). 5.6 Exercise Let P(z) be the predicate "z is a prime number" and the domain of discourse be the set of positive integers Determine the truth value of the following statement: 3r P(r).

Construct a truth table for the proposition and determine whether the statement is a contingency, a tautology, or a contradiction. ал (талр) q -др Ял (алр) T T T Construct the truth table. P T T F T F F FFFF JTF T FFF FFF

Construct a truth table for the compound statement. (p ∧ ~q) ∨ [~(p ∧ q)] p q (p ∧ ~q) ∨ [~ (p ∧ q)] T T  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F F T  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F F F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F  ? T F

Determine whether the logical statement is a tautology, a contradiction or neither: (p→q) → (p v q). [Hint: Use a truth table with 4 rows; you need to check the last row; all T is a tautology: all F is a contradition; a mixture of T and F is neither.] Example statement: If adopting new software leads to savings, then we'll either adopt new software or save money. The parentheses fit in the statement like this: If (adopting new software leads to savings), then (we'll either adopt new software or save money). This interpretation gives adopting new software leads to savings = If we adopt new software, then we'll save money O a. Tautology O b. Neither Oc. Contradiction