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: Name the simple statements.
Let $p$: "$-7$ is an integer" and $q$: "$\sqrt{-7}$ is a complex number".
Step 2: Translate S1.
"If $p$, then $q$" is the conditional $p \rightarrow q$.
Step 3: Translate S2.
"Not $p$ or $q$" is the disjunction $\sim p \lor q$.
Step 4: Recall the key equivalence.
A standard law of logic states $p \rightarrow q \equiv \sim p \lor q$; the conditional and this disjunction always carry the same truth value.
Step 5: Quick truth check.
$p \rightarrow q$ is false only when $p$ is true and $q$ is false; $\sim p \lor q$ is likewise false only in that single case. Their truth columns are identical.
Step 6: Conclude the relationship.
Since S1 and S2 have matching truth tables, they are equivalent statements, which is option 3 and agrees with the key.
\[ \boxed{\text{S1 and S2 are equivalent}} \]