If p and q are logical statements, then `(p^^q)rarr(prarrq)` is equivalent to

Options

A. `p^^q`
B. `prarr(pvvq)`
C. `pvv q`
D. `(pvvq)harr (p^^q)`

Answer

To determine the logical equivalence of the statement \((p \land q) \rightarrow (p \rightarrow q)\), we can use a truth table. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Variables
We have two logical statements, \(p\) and \(q\). The possible truth values for each are True (T) and False (F).

### Step 2: Create a Truth Table
We will create a truth table that includes all combinations of truth values for \(p\) and \(q\).

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

### Step 3: Calculate \(p \land q\)
- \(p \land q\) is True only when both \(p\) and \(q\) are True.
- In our table, it is True only in the first row.

### Step 4: Calculate \(p \rightarrow q\)
- \(p \rightarrow q\) is False only when \(p\) is True and \(q\) is False.
- In our table, it is False in the second row.

### Step 5: Calculate \((p \land q) \rightarrow (p \rightarrow q)\)
- The implication \(A \rightarrow B\) is False only when \(A\) is True and \(B\) is False.
- In our case, since \(p \land q\) is False in all rows except the first, the implication is True in all cases.

### Step 6: Conclusion
The final column shows that \((p \land q) \rightarrow (p \rightarrow q)\) is always True, regardless of the values of \(p\) and \(q\). Therefore, this statement is logically equivalent to True (T).