If p, q and r anre 3 statements, then the truth value of `((-p vv q) ^^ r) implies p` is

Options

A. True if truth values of p, q, r are T, F, T respectively
B. False if truth values of p, q, r are T, F, T respectively
C. False if truth values of p, q, r are T, F, F respectively
D. False if truth values of p, q, r are T, T, T respectively

Answer

To determine the truth value of the statement \((-p \lor q) \land r \implies p\), we will analyze it step by step.

### Step 1: Understand the Components
We have three statements: \(p\), \(q\), and \(r\). The expression can be broken down into its components:
- \(-p\) is the negation of \(p\).
- \(q\) is a statement.
- \(r\) is another statement.

### Step 2: Rewrite the Implication
The implication \(A \implies B\) can be rewritten using logical equivalences:
\[ A \implies B \equiv \neg A \lor B \]
In our case, we can rewrite the original statement as:
\[ \neg((-p \lor q) \land r) \lor p \]

### Step 3: Apply De Morgan's Law
Using De Morgan's laws, we can simplify \(\neg((-p \lor q) \land r)\):
\[ \neg((-p \lor q) \land r) \equiv \neg(-p \lor q) \lor \neg r \]
Now, applying De Morgan's law again on \(\neg(-p \lor q)\):
\[ \neg(-p \lor q) \equiv p \land \neg q \]
Thus, we have:
\[ \neg((-p \lor q) \land r) \equiv (p \land \neg q) \lor \neg r \]

### Step 4: Combine the Expressions
Now we substitute back into our rewritten implication:
\[ (p \land \neg q) \lor \neg r \lor p \]

### Step 5: Analyze the Truth Value
To analyze the truth value, we need to consider all possible truth values for \(p\), \(q\), and \(r\). The expression will be true if at least one of the components is true.

1. If \(p\) is true, then \(p \land \neg q\) can be true regardless of \(q\) and \(r\).
2. If \(p\) is false, we need to check the other components:
- If \(r\) is false, \(\neg r\) is true, making the whole expression true.
- If \(q\) is true, then \(\neg q\) is false, and we need \(r\) to be false for the expression to hold.

### Conclusion
The truth value of the statement \((-p \lor q) \land r \implies p\) can be true or false depending on the values of \(p\), \(q\), and \(r\).