Question

Which one of the following is false (a) p \(\Lambda(\sim \mathrm{p})\) is a contradiction (b) \((\mathrm{p} \Rightarrow \mathrm{q}) \Leftrightarrow(\sim \mathrm{q} \Rightarrow \sim \mathrm{p})\) is a contradiction (c) \(\sim(\sim p) \Leftrightarrow p\) is a tautology (d) \(p \mathrm{~V}(\sim \mathrm{p})\) is a tautology

Short answer

(b) $(p \Rightarrow q) \Leftrightarrow(\sim q \Rightarrow \sim p)$ is false, as it is not a contradiction.

Step-by-step solution

Check Statement (a): p ∧(~p) is a contradiction

In this statement, we need to check if p and its negation (~p) can both be true at the same time. Recall that a contradiction is a statement that is always false, regardless of the truth values of its variables. Let's create a truth table for this expression: | p | ~p | p ∧(~p) | |---|----|---------| | T | F | F | | F | T | F | As we can see from the truth table, p ∧(~p) is always false, regardless of the value of p. Therefore, statement (a) is indeed a contradiction.

Check Statement (b): (p → q) ↔ (~q → ~p) is a contradiction

In this statement, we need to determine if the biconditional expression relating the logical implications (p → q) and (~q → ~p) is a contradiction. First, let's recall that the logical implication (p → q) is equivalent to (~p ∨ q). So we can rewrite the biconditional expression as follows: (~p ∨ q) ↔ (p ∨ ~q) Next, let's create a truth table for this expression: | p | q | ~p | ~q | ~p ∨ q | p ∨ ~q | (~p ∨ q) ↔ (p ∨ ~q) | |---|---|----|----|--------|--------|---------------------| | T | T | F | F | T | T | T | | T | F | F | T | F | T | F | | F | T | T | F | T | F | F | | F | F | T | T | T | T | T | The biconditional expression is not always false, which means that statement (b) is not a contradiction.

Check Statement (c): ~(~p) ↔ p is a tautology

In this statement, we have the negation of the negation of p, which is simply p. The statement ~(~p) ↔ p is asking whether the double negation of p is equivalent to p itself. Let's create a truth table for this expression: | p | ~p | ~(~p) | ~(~p) ↔ p | |---|----|-------|----------| | T | F | T | T | | F | T | F | T | The biconditional expression is always true, which means that statement (c) is indeed a tautology.

Check Statement (d): p ∨(~p) is a tautology

In this statement, we have a disjunction between p and its negation (~p). Let's create a truth table for this expression: | p | ~p | p ∨(~p) | |---|----|---------| | T | F | T | | F | T | T | The expression p ∨(~p) is always true, which means that statement (d) is indeed a tautology. Based on our analysis, statement (b) is the only one that is false, and all other statements are true. So the correct answer is (b).

Key concepts

Contradiction

A contradiction is a type of statement that is always false, no matter the truth values of its components. This means that no combination of truth values will make the statement true.
To identify a contradiction, a truth table is a handy tool. For instance, when checking the statement \( p \land (\sim p) \), meaning 'p and not p', the contradiction becomes evident.

• When \( p \) is true, \( \sim p \) is false, hence \( p \land (\sim p) \) equals false.
• Similarly, when \( p \) is false, \( \sim p \) is true, but \( p \land (\sim p) \) remains false.

So, as illustrated in the solution, the statement \( p \land (\sim p) \) is always false, confirming it as a contradiction.

Tautology

A tautology is a logical statement that is always true, regardless of the truth values of its variables. It's the opposite of a contradiction.
For example, the expression \( p \lor (\sim p) \), which translates to 'p or not p', is a classic tautology.

• When \( p \) is true, even though \( \sim p \) is false, \( p \lor (\sim p) \) is true.
• Conversely, when \( p \) is false, \( \sim p \) is true, thus \( p \lor (\sim p) \) still remains true.

This continuous truth across all situations confirms that \( p \lor (\sim p) \) is indeed a tautology.

Truth Table

Truth tables are fundamental tools in logic to systematically examine the truth values of logical expressions based on all possible combinations of truth values.
In constructing a truth table:

• List all variables and their possible true (T) or false (F) values on separate columns.
• Include columns for each component of the given logical statement.
• Determine and fill in the resulting truth values for the overall expression by applying logical operations.

Truth tables help us identify whether a statement is a contradiction, a tautology, or neither by showing the truth value for each possibility.
In the solution, truth tables were created for different statements to verify their nature, allowing us to conclude whether they were contradictions or tautologies.

Logical Implications

Logical implications are statements of the form \( p \Rightarrow q \) meaning 'if p then q'. This concept is critical in understanding conditionals in logic.
The implication \( p \Rightarrow q \) can be rewritten with the logical OR operator as \( \sim p \lor q \), establishing that \( q \) is true whenever \( p \) is true, or \( p \) is false.
With logical implications:

• The statement is false only when \( p \) is true and \( q \) is false.
• In all other cases, the statement is considered true.

By using truth tables, we check biconditional expressions like \( (p \Rightarrow q) \Leftrightarrow (\sim q \Rightarrow \sim p) \).
This helps determine if such implications are equivalences, tautologies, contradictions, or something else.