Determine whether the statement is a contradiction -

Question

Q1 Use a truth table to determine whether the following statement is a contradiction, a tautology or neither. If it is a contradiction or a tautology, verify your answer using logical equivalences.
((p → q) ∧ (r → ∼ q)) → (r → ∼ p)

Q2 Show that the following argument with hypotheses on lines 1-3 and conclusion on line c is valid, by supplementing steps using the rules of inference (Table 2.3.1) and/or logical equivalences (Theorem 2.1.1). Clearly label which rule you used in each step.

1. p → q
2. ∼ (q ∧ r)
3. r
c. ∼ p

Q3 Express each of the following statements using only the symbols p q ∧ ∼ ( ):
(a) p ∨ q
(b) p → q
(c) p ⊕ q
Justify your answers, using either logical equivalences or truth tables.

Q4 Let P (x), Q(x), R(x) and S(x) denote the following predicates with domain Z:
P (x): x² = x³,
Q(x): x ≥ 0,
R(x): x² < 0,
S(x): x is odd.

(a) For each predicate, determine its truth set.

(b) Determine whether each of the following statements is true or false, and give reasons.

∀x ∈ Z, P (x) → Q(x) (1)

∀x ∈ Z, Q(x) → P (x) (2)

∀x ∈ Z, R(x) → S(x) (3)

∀x ∈ Z, S(x) → R(x) (4)

∃x ∈ Z such that Q(x) ∧ S(x) (5)

(c) Write down the negation of each statement in part (b).

(>d) Determine whether each of the following statements is true or false, and give reasons.

∃y ∈ Z such that ∀x ∈ Z, Q(x + y) (6)

∀x ∈ Z, ∃y ∈ Z such that Q(x + y) (7)

(>e) Write down the negation of each statement in part (d).