Problem 19 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The truth table is [[T, T, F], [T, F, T], [F, T, T], [F, F, F]], where the first, second, and third columns represent \( p \), \( q \), and \( \sim(p \leftrightarrow q) \) respectively.

Step 1: Understand logical operations

First, you need to understand the basic logical operations involved in the exercise. The biconditional operator \( p \leftrightarrow q \), also known as 'if and only if', is true when both \( p \) and \( q \) have same truth values. Either both are true, or both are false. The negation operator \( \sim \) flips the truth value, turning true to false and vice versa.

Step 2: Create the basic truth table for p and q

A truth table with two variables \( p \) and \( q \) has four possible combinations of truth values - (T,T), (T,F), (F,T) and (F,F). So, list down these combinations.

Step 3: Determine the values of \( p \leftrightarrow q \)

Next, determine the values of \( p \leftrightarrow q \) for each combination. This is true when both \( p \) and \( q \) have same truth values. Either both are true, or both are false.

Step 4: Determine the values of \( \sim(p \leftrightarrow q) \)

Lastly, determine the values of \( \sim(p \leftrightarrow q) \) for every combination by negating the values of \( p \leftrightarrow q \).

Key Concepts

Logic Operators

Logic operators are the fundamental building blocks in logic. They manipulate propositions or statements to produce new outcomes. In logical expressions, there are various types of operators that help define relationships between statements, conditions, or values. Some common logic operators include:

• AND (\(\wedge\)
• OR (\(\vee\)
• NOT (negation, \(\sim\)
• IMPLIES (\(\rightarrow\)
• BICONDITIONAL (\(\leftrightarrow\)

Each operator has specific rules governing how it interacts with truth values. These rules form a critical part of constructing and interpreting truth tables, which visually represent the outcomes of logical expressions.

Biconditional Operator

The biconditional operator is a specific type of logic operator denoted by \(p \leftrightarrow q\).It implies "if and only if" and is true in two main scenarios:

• Both propositions have the same truth value, i.e., both \(p\) and \(q\) are true.
• Both propositions are false.

This means, if both are either true or false, then the outcome of \(p \leftrightarrow q\) is true.
When their truth values differ, the result is false. This operator is helpful for expressions that establish a strong form of equality between two propositions.

Negation Operator

The negation operator flips the truth value of any statement.
It is represented by the symbol \(\sim\).When applied to a proposition, it transforms:

• True onto false
• False onto true

This operator's role is to invert the output of a statement, thus altering the logical relationship.
For instance, in the expression \(\sim(p \leftrightarrow q)\), after determining the outcome of the biconditional \(p \leftrightarrow q\), use the negation operator to flip its truth value.
This step is crucial to fully interpret the given logical statement.

Truth Values

Truth values are essential components in evaluating logical expressions and form the basis of truth tables.
They signify the validity of a statement, represented as either TRUE (T) or FALSE (F).
In a truth table, all possible combinations of truth values for the involved variables are listed and evaluated.

• Truth value "TRUE" is represented as T.
• Truth value "FALSE" is represented as F.

For a statement involving variables like \(p\) and \(q\), possible combinations include (T, T), (T, F), (F, T), and (F, F).
Each row in the truth table represents a different scenario and helps in understanding the behavior of the logical expression under different conditions.