Question

S1 : If $-7$ is an integer, then $\sqrt{-7}$ is a complex number
S2 : $-7$ is not an integer or $\sqrt{-7}$ is a complex number
Choose the correct statement from the options given below

• A. S1 and S2 are converse statements of each other
• B. S1 and S2 are negations of each other
• C. S1 and S2 are equivalent statements
• D. S1 and S2 are contrapositive of each other

Hint:

To remember this fundamental logical rule easily, think of the conditional arrow $p \rightarrow q$ as "breaking" into a negation on the left side and turning the arrow into an "OR" wedge ($\sim p \lor q$). They are identical in every mathematical proof!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem presents two statements, S1 and S2, expressed in words. We need to determine the logical relationship between them by converting them into symbolic forms.

Step 2: Key Formula or Approach:
Let us define the simple component propositions: $p: -7$ is an integer. $q: \sqrt{-7}$ is a complex number. The conditional statement "If $p$, then $q$" is symbolically written as $p \rightarrow q$.
The disjunctive statement "Not $p$ or $q$" is symbolically written as $\sim p \lor q$.
A well-known logical equivalence in mathematical logic states that: $$p \rightarrow q \equiv \sim p \lor q$$

Step 3: Detailed Explanation:
1. Translate statement S1 into its logical form: "If $-7$ is an integer, then $\sqrt{-7}$ is a complex number" corresponds directly to the conditional form: $$\text{S1} \equiv p \rightarrow q$$ 2. Translate statement S2 into its logical form: "$-7$ is not an integer or $\sqrt{-7}$ is a complex number" corresponds directly to the disjunctive form: $$\text{S2} \equiv \sim p \lor q$$ 3. Let us verify their truth values using a truth table analysis or known conditional laws. The conditional operator $p \rightarrow q$ is defined to be false only when the antecedent $p$ is true and the consequent $q$ is false. In all other cases, it is true. The expression $\sim p \lor q$ yields the exact same truth outputs for all possible combinations of $p$ and $q$.
Therefore, because $p \rightarrow q \equiv \sim p \lor q$, the statements S1 and S2 are logically equivalent.

Step 4: Final Answer:
S1 and S2 are equivalent statements, which corresponds to option (C).