Verify whether each of the following statements is a tautology or contradiction: iv) (A B) ( A B) v) (A B) ( A B)

Okay, I will provide the remaining solutions.

2) Verify whether each of the following statements is a tautology or contradiction:

• iv) $\sim (A \lor B) \Leftrightarrow (\sim A \land \sim B)$ This statement is a tautology.
  • Explanation: This is De Morgan's Law, which states that the negation of a disjunction is logically equivalent to the conjunction of the negations. We can verify this with a truth table:

$A$	$B$	$A \lor B$	$\sim (A \lor B)$	$\sim A$	$\sim B$	$\sim A \land \sim B$	$\sim (A \lor B) \Leftrightarrow (\sim A \land \sim B)$
T	T	T	F	F	F	F	T
T	F	T	F	F	T	F	T
F	T	T	F	T	F	F	T
F	F	F	T	T	T	T	T

• v) $(A \lor B) \land (\sim A \land \sim B)$ This statement is a contradiction.
  • Explanation: $$(A \lor B) \land (\sim A \land \sim B)$$ Using De Morgan's Law, $(\sim A \land \sim B) \equiv \sim (A \lor B)$: $$(A \lor B) \land \sim (A \lor B)$$ Let $P = (A \lor B)$. Then the expression becomes $P \land \sim P$. By the Complement Law, $P \land \sim P$ is always False. $$\boxed{F}$$ The statement is always false.

5) Consider the conditional statement;
"If I must withdraw with my debit card, then I must remember my PIN"

Let $P$ be "I must withdraw with my debit card".
Let $Q$ be "I must remember my PIN".
The original statement is $P \Rightarrow Q$.

• c. Contra-positive The contra-positive of $P \Rightarrow Q$ is $\sim Q \Rightarrow \sim P$.
  • Statement: "If I do not remember my PIN, then I must not withdraw with my debit card."