Question

Let \(p, q, r \)be three logical statements. Consider the compound statements
\(S_1 : ((~p) ∨q) ∨ ((~p) ∨r)\) and
\(S_2 :p→ (q∨r)\)
Then, which of the following is NOT true?

• A. If \(S_2\) is True, then \(S_1\)is True
• B. If \(S_2\) is False, then \(S_1\) is False
• C. If \(S_2\) is False, then \(S_1\) is True
• D. If \(S_1\) is False, then \(S_2\) is False

Answer 1

The Correct Option is C

To determine which statements regarding \(S_1\) and \(S_2\) are not true, let's analyze both logical expressions:

Given logical statements:

\(S_1 : ((\sim p) \lor q) \lor ((\sim p) \lor r)\)

\(S_2 : p \rightarrow (q \lor r)\)

Firstly, let's simplify \(S_1\):

1. \((\sim p) \lor q\) is True if \(\sim p\) is True or \(q\) is True.
2. \((\sim p) \lor r\) is True if \(\sim p\) is True or \(r\) is True.

The expression \(S_1\): \(((\sim p) \lor q) \lor ((\sim p) \lor r)\) can be further simplified. Since both disjunctions contain \(\sim p\), the overall expression simplifies to:

\((\sim p) \lor (q \lor r)\)

Now, considering \(S_2\): \(p \rightarrow (q \lor r)\) is logically equivalent to \(\sim p \lor (q \lor r)\) because an implication \(a \rightarrow b\) is the same as \(\sim a \lor b\).

So we see that:

1. \(S_1 : (\sim p) \lor (q \lor r)\)
2. \(S_2 : (\sim p) \lor (q \lor r)\)

Both \(S_1\) and \(S_2\) are logically equivalent since they simplify to the same expression.

Now let's verify each option based on this understanding:

1. If \(S_2\) is True, then \(S_1\) is True: This is correct since they are logically equivalent.
2. If \(S_2\) is False, then \(S_1\) is False: This is also correct because if one expression is False, the other must be False since they are equivalent.
3. If \(S_2\) is False, then \(S_1\) is True: This is not true because the logical equivalence indicates that both would be False.
4. If \(S_1\) is False, then \(S_2\) is False: Correct, as they are equivalent expressions.

Therefore, the statement that is NOT true is: If \(S_2\) is False, then \(S_1\) is True.