Problem 8 - Thinking Mathematically

Question

Construct a truth table for the given statement. \((p \rightarrow q) \wedge \sim p\)

Step-by-Step Solution

Answer

The truth table for the given statement \((p \rightarrow q) \wedge \sim p\) is as follows:| p | q | \(p \rightarrow q\) | \(\sim p\) | \((p \rightarrow q) \wedge \sim p\) || :-: | :-: | :-----------------: | :------: | :---------------------------: || T | T | T | F | F || T | F | F | F | F || F | T | T | T | T || F | F | T | T | T |Here, T represents true and F represents false.

Step 1: Define the possible input values

Before creating the truth table, recognize that each of the variables \(p\) and \(q\) can either be true (T) or false (F). This results in 4 possible combinations of inputs for our statement: TT, TF, FT, FF.

Step 2: Fill in Values for \(p \rightarrow q\)

Next, fill in truth values for the statement \(p \rightarrow q\) using the rule of implications. The implication \(p \rightarrow q\) is false if and only if \(p\) is true and \(q\) is false, otherwise it is true. According to our input combinations: if \(p\) and \(q\) are T (true), \(p \rightarrow q\) will be T (true). If \(p\) is T (true) and \(q\) is F (false), \(p \rightarrow q\) becomes F (false). Now, if \(p\) is F(false), according to the rule, \(p \rightarrow q\) becomes T (true), no matter what the value of \(q\) is.

Step 3: Provide Values for \(\sim p\)

Once we've filled in the values for \(p \rightarrow q\), next, we find values for the negation of \(p\) or \(\sim p\). The negation of a statement is simply the opposite of its truth value. So, if \(p\) is T (true), \(\sim p\) is F (false), and vise-versa.

Step 4: Determine values for the overall statement \((p \rightarrow q) \wedge \sim p\)

After calculating \(p \rightarrow q\) and \(\sim p\), now, apply the conjunction operation (\(\wedge\)). The conjunction of two statements is true if and only if both statements are true. So for any combination where either \(p \rightarrow q\) or \(\sim p\) is F (false), the overall statement \((p \rightarrow q) \wedge \sim p\) is F (false). The overall statement is true if and only if both \(p \rightarrow q\) and \(\sim p\) are T (true).

Key Concepts

Implication in Logic

In logic, an implication is a conditional statement connecting two propositions. This connection is often presented in the form \( p \rightarrow q \) and can be read as "if \( p \), then \( q \)." Here, \( p \) is the antecedent, and \( q \) is the consequent. The truth of the entire implication is based on the relationship between \( p \) and \( q \).

• If both \( p \) and \( q \) are true, the implication \( p \rightarrow q \) is true.
• If \( p \) is true and \( q \) is false, the implication \( p \rightarrow q \) is false.
• If \( p \) is false, the implication \( p \rightarrow q \) is true regardless of \( q \).

This may seem counterintuitive at first, especially the last rule, but think of it this way: an implication \( p \rightarrow q \) does not make any claims on \( q \) when \( p \) is false. Logic focuses on the cases when \( p \) is true and does not affect \( q \). Understanding this is essential for constructing accurate truth tables involving implications.

Negation in Logic

Negation is a simple but powerful concept in logic. It involves flipping the truth value of a statement to its opposite. For example, if \( p \) is true, then its negation \( \sim p \) is false, and vice versa.

The symbol \( \sim \) represents negation and is always positioned in front of the statement it negates. This operation is crucial for analyzing logical statements, especially in combination with other logical connectives like conjunction or implication.

In a truth table, negation simply alters the column under the proposition for which the negation is applied. By understanding and utilizing negation, you can explore alternative outcomes in logical expressions and equations.

Conjunction in Logic

Conjunction is a fundamental logical operation indicated by the symbol \( \wedge \). It is used to connect two propositions \( p \) and \( q \), forming a new statement \( p \wedge q \) which is true only if both \( p \) and \( q \) are true simultaneously.

• If both \( p \) and \( q \) are true, then \( p \wedge q \) is true.
• If either \( p \) or \( q \) is false, then \( p \wedge q \) is false.

This binary operation mimics the "and" condition commonly used in everyday language. It is important in constructing composite logical statements and evaluating their truthfulness within truth tables. By applying the rules of conjunction, one can determine how multiple conditions affect overall statements in logic.