Determine the truth value of the following:
1. (p ∧ q) → (p ∧ q)
2. ¬p⋁(p → q)
3. (p → ¬q) ↔ (¬p → q)
To determine the truth value of each statement, we can verify the truth value of each component within the statement.
1. (p ∧ q) → (p ∧ q)
In this statement, we have a conditional statement (→) with the antecedent (p ∧ q) and the consequent (p ∧ q).
To determine the truth value of a conditional statement, we need to check the truth values of the antecedent and the consequent. Since (p ∧ q) is true when both p and q are true, and (p ∧ q) is always true, irrespective of the truth values of p and q. Thus, the conditional statement is always true.
2. ¬p ⋁ (p → q)
This statement has a disjunction (⋁) of two components: ¬p and (p → q).
To determine the truth value of a disjunction, we need to check the truth values of each component. If either component is true, then the disjunction is also true.
The component ¬p is true when p is false.
The component (p → q) is true when p is false or q is true.
So, we have two possible cases:
- If p is false and q is true, then ¬p ⋁ (p → q) is true.
- If p is false and q is false or p is true and q is true, then ¬p ⋁ (p → q) is true.
Therefore, ¬p ⋁ (p → q) is always true.
3. (p → ¬q) ↔ (¬p → q)
This statement has a biconditional (↔) relation between two implications: (p → ¬q) and (¬p → q).
To determine the truth value of a biconditional, we need to check the truth values of both implications.
The component (p → ¬q) is true when either p is false or q is false.
The component (¬p → q) is true when either ¬p is false (i.e., p is true) or q is true.
So, we have three possible cases:
- If p is false and q is false, then (p → ¬q) ↔ (¬p → q) is true.
- If p is true and q is false or p is false and q is true, then (p → ¬q) ↔ (¬p → q) is false.
- If p is true and q is true, then (p → ¬q) ↔ (¬p → q) is true.
Therefore, (p → ¬q) ↔ (¬p → q) is not always true and depends on the specific truth values of p and q.