Question

Consider the following three statements:
(A) If $3 + 2 = 7$ then $4 + 3 = 8$.
(B) If $5 + 2 = 7$ then earth is flat.
(C) If both (A) and (B) are true then $5 + 6 = 11$.
Which of the following statements is correct?

• (A) and (C) are true while (B) is false.
• (A) is true while (B) and (C) are false.
• (A) is false but (B) and (C) are true.
• (A) is false while (C) is true.

Hint:

In formal logic, a conditional statement is vacuously true whenever the "If" part (the premise) is false. If you start with a lie, you can logically conclude absolutely anything! (False $\rightarrow$ True = True, and False $\rightarrow$ False = True).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given three conditional logic statements ("If $p$, then $q$"). We must evaluate the truth value (True or False) of each composite statement based on the rules of mathematical logic.

Step 2: Key Formula or Approach:
The truth value of a conditional statement $p \rightarrow q$ follows this exact truth table:
- $T \rightarrow T \equiv T$
- $T \rightarrow F \equiv F$ (This is the ONLY scenario where a conditional is false)
- $F \rightarrow T \equiv T$
- $F \rightarrow F \equiv T$

Step 3: Detailed Explanation:
Evaluate statement (A): "If $3 + 2 = 7$ then $4 + 3 = 8$."
- Premise $p$: $3+2=7$ is False (F).
- Conclusion $q$: $4+3=8$ is False (F).
- Structure: $F \rightarrow F$. According to logic rules, this makes statement (A) True.
Evaluate statement (B): "If $5 + 2 = 7$ then earth is flat."
- Premise $p$: $5+2=7$ is True (T).
- Conclusion $q$: "earth is flat" is False (F).
- Structure: $T \rightarrow F$. According to logic rules, this makes statement (B) False.
Evaluate statement (C): "If both (A) and (B) are true then $5 + 6 = 11$."
- Premise $p$: "Both (A) and (B) are true". Since (B) is False, their logical AND is False (F).
- Conclusion $q$: $5+6=11$ is True (T).
- Structure: $F \rightarrow T$. According to logic rules, this makes statement (C) True.
Summary: (A) is True, (B) is False, (C) is True.

Step 4: Final Answer:
"(A) and (C) are true while (B) is false" corresponds to option (a).