If p, q and r are three logical statements then the truth value of the statement `(p^^~q)vv(qrarr r)`, where p is true, is

Options

A. True if q is true
B. False if q is true
C. True if q is false
D. False is q is false

Answer

To solve the logical statement `(p ∧ ¬q) ∨ (q → r)` where `p` is true, we can follow these steps:

### Step 1: Identify the values of p, q, and r
We know that:
- `p` is true.
- We will consider two cases for `q`: when `q` is true and when `q` is false.
- The value of `r` is not specified, so we will analyze how it affects the overall expression.

### Step 2: Rewrite the implication
The implication `q → r` can be rewritten using logical equivalence:
- `q → r` is equivalent to `¬q ∨ r`.

Thus, our expression becomes:
`(p ∧ ¬q) ∨ (¬q ∨ r)`

### Step 3: Analyze case 1: When q is true
If `q` is true:
- `¬q` (negation of q) is false.
- The expression simplifies to:
- `(true ∧ false) ∨ (false ∨ r)`
- This further simplifies to:
- `false ∨ (false ∨ r)`, which is `false ∨ r`.

Since `r` can be either true or false, the overall expression will depend on `r`. Therefore, if `q` is true, the expression can be either true or false.

### Step 4: Analyze case 2: When q is false
If `q` is false:
- `¬q` is true.
- The expression simplifies to:
- `(true ∧ true) ∨ (true ∨ r)`
- This further simplifies to:
- `true ∨ (true ∨ r)`, which is `true`.

In this case, regardless of the value of `r`, the overall expression is true.

### Conclusion
- When `q` is true, the truth value of the expression depends on `r`.
- When `q` is false, the truth value of the expression is true.

Thus, the truth value of the statement `(p ∧ ¬q) ∨ (q → r)` is true if `q` is false.

### Final Answer
The correct option is **C: true if Q is false**.

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