If a compound statement r is contradiction , then the truth value of ` ( p Rightarrow q) ^^ r ^^ p [ p Rightarrow ~ r ] ` is

Options

A. T
B. F
C. T or F
D. none of these

Answer

To solve the problem, we need to analyze the given compound statement step by step. The statement is:

`(p ⇒ q) ∧ r ∧ (p ⇒ ¬r)`

We are given that `r` is a contradiction, which means `r` is always false.

### Step 1: Understand the implications of `r` being a contradiction
Since `r` is a contradiction, we can replace `r` with `False`.

### Step 2: Rewrite the statement
Now, we can rewrite the original statement with `r` replaced by `False`:

`(p ⇒ q) ∧ False ∧ (p ⇒ ¬False)`

### Step 3: Simplify the statement
Next, we simplify the expression. The expression `(p ⇒ ¬False)` simplifies to `(p ⇒ True)`. The implication `p ⇒ True` is always true regardless of the truth value of `p`.

So, we can rewrite the statement as:

`(p ⇒ q) ∧ False ∧ True`

### Step 4: Analyze the conjunction
Now we have:

`(p ⇒ q) ∧ False ∧ True`

According to the rules of conjunction (AND operation), if any part of the conjunction is false, the entire expression evaluates to false. Since we have `False` in our expression, we can conclude that:

`(p ⇒ q) ∧ False ∧ True = False`

### Final Result
Thus, the truth value of the entire compound statement is `False`.