Relations & Functions
Properties of relation; domain/range of inverse
Grade Class 12
Question:
Let $R=\{(a,b): a/b$ is a prime number$\}$ on first twenty natural numbers. Consider: (I) $R$ is reflexive and symmetric but not transitive; (II) Range of $R^{-1}$ has 20 elements; (III) Domain of $R^{-1}$ has 10 elements. Which statements are true?
I, II, III
II, III
II
III
Step-by-Step Solution
Key Concept: $a/b$ is prime: $R$ is neither reflexive ($1/1=1$ not prime) nor symmetric ($6/2=3$ prime but $2/6$ not integer). Domain of $R$ = $\{b:\exists a,a/b$ prime$\}$ = $\{1,2,...,10\}$ (divisors). Range = $\{a:a/b$ prime for some $b\}$. Domain of $R^{-1}$ = range of $R$.
Only (III) is true.
Correct Answer: 4