Construct truth tables for the following :
`(p implies q)^^(qimpliesp)`

Answer

To construct the truth table for the expression \( (p \implies q) \land (q \implies p) \), we will follow these steps:

### Step 1: Identify the variables
We have two variables: \( p \) and \( q \).

### Step 2: List all possible truth values
The truth values for \( p \) and \( q \) can be either True (T) or False (F). Therefore, we have the following combinations:

1. \( p = T, q = T \)
2. \( p = T, q = F \)
3. \( p = F, q = T \)
4. \( p = F, q = F \)

### Step 3: Create the truth table structure
We will create a table with the following columns:
- Column 1: \( p \)
- Column 2: \( q \)
- Column 3: \( p \implies q \)
- Column 4: \( q \implies p \)
- Column 5: \( (p \implies q) \land (q \implies p) \)

### Step 4: Fill in the truth values for \( p \implies q \)
The implication \( p \implies q \) is defined as follows:
- True if both \( p \) and \( q \) are true.
- False if \( p \) is true and \( q \) is false.
- True if \( p \) is false (regardless of \( q \)).

Now, we fill in the values for \( p \implies q \):

1. \( p = T, q = T \) → \( p \implies q = T \)
2. \( p = T, q = F \) → \( p \implies q = F \)
3. \( p = F, q = T \) → \( p \implies q = T \)
4. \( p = F, q = F \) → \( p \implies q = T \)

### Step 5: Fill in the truth values for \( q \implies p \)
The implication \( q \implies p \) is defined similarly:
- True if both \( q \) and \( p \) are true.
- False if \( q \) is true and \( p \) is false.
- True if \( q \) is false (regardless of \( p \)).

Now, we fill in the values for \( q \implies p \):

1. \( p = T, q = T \) → \( q \implies p = T \)
2. \( p = T, q = F \) → \( q \implies p = T \)
3. \( p = F, q = T \) → \( q \implies p = F \)
4. \( p = F, q = F \) → \( q \implies p = T \)

### Step 6: Fill in the final column for \( (p \implies q) \land (q \implies p) \)
Now we will perform the AND operation between the two previous columns:

1. \( T \land T = T \)
2. \( F \land T = F \)
3. \( T \land F = F \)
4. \( T \land T = T \)

### Final Truth Table
Now we can summarize our truth table:

| \( p \) | \( q \) | \( p \implies q \) | \( q \implies p \) | \( (p \implies q) \land (q \implies p) \) |
|---------|---------|---------------------|---------------------|--------------------------------------------|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |