Greene Grant
06/17/2023 · Junior High School
Construct a truth table for the statement \( (q \rightarrow r) \leftrightarrow \sim p \).
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Step-by-step Solution
To construct a truth table for the statement \( (q \rightarrow r) \leftrightarrow \sim p \), we need to evaluate the components of the expression step by step. The components involved are \( p \), \( q \), and \( r \), as well as the logical operations of implication (\( \rightarrow \)), negation (\( \sim \)), and biconditional (\( \leftrightarrow \)).
### Step 1: Identify the variables
We have three variables: \( p \), \( q \), and \( r \).
### Step 2: Determine the number of rows
Since there are three variables, we will have \( 2^3 = 8 \) rows in our truth table.
### Step 3: Create the truth table
We will calculate the values for \( q \rightarrow r \), \( \sim p \), and finally \( (q \rightarrow r) \leftrightarrow \sim p \).
| \( p \) | \( q \) | \( r \) | \( q \rightarrow r \) | \( \sim p \) | \( (q \rightarrow r) \leftrightarrow \sim p \) |
|---------|---------|---------|-----------------------|---------------|-----------------------------------------------|
| T | T | T | T | F | F |
| T | T | F | F | F | T |
| T | F | T | T | F | F |
| T | F | F | T | F | F |
| F | T | T | T | T | T |
| F | T | F | F | T | F |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
### Step 4: Fill in the columns
1. **Calculate \( q \rightarrow r \)**:
- \( q \rightarrow r \) is true unless \( q \) is true and \( r \) is false.
2. **Calculate \( \sim p \)**:
- \( \sim p \) is true when \( p \) is false.
3. **Calculate \( (q \rightarrow r) \leftrightarrow \sim p \)**:
- This is true when both sides are the same (both true or both false).
### Final Truth Table
The final truth table is as follows:
| \( p \) | \( q \) | \( r \) | \( q \rightarrow r \) | \( \sim p \) | \( (q \rightarrow r) \leftrightarrow \sim p \) |
|---------|---------|---------|-----------------------|---------------|-----------------------------------------------|
| T | T | T | T | F | F |
| T | T | F | F | F | T |
| T | F | T | T | F | F |
| T | F | F | T | F | F |
| F | T | T | T | T | T |
| F | T | F | F | T | F |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
This table summarizes the truth values for the expression \( (q \rightarrow r) \leftrightarrow \sim p \) based on all possible combinations of truth values for \( p \), \( q \), and \( r \).
Quick Answer
The truth table for \( (q \rightarrow r) \leftrightarrow \sim p \) is as follows:
| \( p \) | \( q \) | \( r \) | \( q \rightarrow r \) | \( \sim p \) | \( (q \rightarrow r) \leftrightarrow \sim p \) |
|---------|---------|---------|-----------------------|---------------|-----------------------------------------------|
| T | T | T | T | F | F |
| T | T | F | F | F | T |
| T | F | T | T | F | F |
| T | F | F | T | F | F |
| F | T | T | T | T | T |
| F | T | F | F | T | F |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
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