Question

Statement I Any subset of $R$ that is an inductive set must contain $N$.
Statement II A set $S$ is said to be an inductive set, if $1\in S$ and $x+1\in S$, whenever $x\in S$.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (A)

We know, the set of natural numbers $N$ is a special ordered subset of the real numbers. Infact, $N$ is the smallest subset of $R$ with the following property.
A set $S$ is said to be an inductive set, if $1\in S$ and $x+1\in S$ whenever $x\in S$. Since, $N$ is the smallest subset of $R$ which is an inductive set, it follows that any subset of $R$ that is an inductive set must contain $N$.
So, both the statements are true and Statement II is the correct explanation of Statement I.