a) By use of truth tables, establish whether the statements given below are a tantology or contradiction (i) (pwedge q)vee (neg pvee (pwedge neg q) (5 marks) (ii) . (pwedge neg q)wedge (neg pvee q) (5 marks) b) Use a truth table to establish whether the argument below is valid. pwedge qarrow neg r pVneg q

a) (i) To determine whether the statement is a tautology or contradiction, we can construct a truth table:| p | q | p ∧ q | ¬p | p ∧ ¬q | ¬p ∨ (p ∧ ¬q) | (p ∧ q) ∨ (¬p ∨ (p ∧ ¬q)) ||---|---|-------|----|-------|-------------|---------------------|| T | T | T | F | F | F | T || T | F | F | F | F | F | F || F | T | F | T | F | F | F || F | F | F | T | T | T | T |From the truth table, we can see that the statement is a tautology because it is true for all possible combinations of truth values for p and q.(ii) To determine whether the statement is a tautology or contradiction, we can construct a truth table:| p | q | ¬q | p ∧ ¬q | ¬p | ¬p ∨ q | (p ∧ ¬q) ∧ (¬p ∨ q) ||---|---|----|-------|----|-------|------------------|| T | T | F | F | F | T | F || T | F | T | T | F | F | F || F | T | F | F | T | T | T || F | F | T | F | T | T | T |From the truth table, we can see that the statement is neither a tautology nor a contradiction because it is true for some combinations of truth values for p and q.b) To determine whether the argument and is valid, we can construct a truth table:| p | q | r | ¬r | p ∧ q | ¬q | p ∧ q → ¬r | p ∨ ¬q | (p ∧ q → ¬r) ∧ (p ∨ ¬q) ||---|---|---|----|-------|----|-----------|-------|---------------------|| T | T | T | F | T | F | F | T | F || T | T | F | T | T | F | T | T | T || T | F | T | F | F | T | F | T | F || T | F | F | T | F | T | T | T | T || F | T | T | F | F | F | F | F | F || F | T | F | T | F | F | T | F | F || F | F | T | F | F | T | F | T | F || F | F | F | T | F | T | T | T | T |From the truth table, we can see that the argument and is valid because the compound statement is true for all possible combinations of truth values for p, q, and r.