.If `p,q` and `r` are three propositions,then which of the following combination of truth values of `p,q` and `r` makes the logical expression `{(p vv q)^^((~p) vv r} to ((~q) vv r)` false ?

Options

A. `p=T,q=T,r=F`
B. `p=T,q=F,r=T`
C. `p=F,q=T,r=F`
D. `p=T,q=F,r=F,r=F`

Answer

To solve the logical expression \((p \lor q) \land (\neg p \lor r) \implies (\neg q \lor r)\) and determine the combination of truth values of \(p\), \(q\), and \(r\) that makes it false, we can follow these steps:

### Step 1: Understand the Implication
The expression \(A \implies B\) is false only when \(A\) is true and \(B\) is false. Thus, we need to find when:
1. \(A = (p \lor q) \land (\neg p \lor r)\) is true.
2. \(B = (\neg q \lor r)\) is false.

### Step 2: Analyze \(B\)
For \(B = (\neg q \lor r)\) to be false, both \(\neg q\) and \(r\) must be false:
- \(\neg q\) is false when \(q\) is true.
- \(r\) is false.

Thus, we have:
- \(q = T\) (True)
- \(r = F\) (False)

### Step 3: Analyze \(A\)
Now we need to check when \(A = (p \lor q) \land (\neg p \lor r)\) is true given \(q = T\) and \(r = F\).

1. **Evaluate \(p \lor q\)**:
- Since \(q\) is true, \(p \lor q\) will be true regardless of \(p\).

2. **Evaluate \(\neg p \lor r\)**:
- Since \(r\) is false, \(\neg p \lor r\) will be true only if \(\neg p\) is true, which means \(p\) must be false.

### Step 4: Determine Values
From the analysis:
- For \(A\) to be true, we need:
- \(p = F\) (False)
- \(q = T\) (True)
- \(r = F\) (False)

### Conclusion
The combination of truth values that makes the logical expression false is:
- \(p = F\)
- \(q = T\)
- \(r = F\)

### Final Answer
Thus, the answer is:
- \(p = F\), \(q = T\), \(r = F\)