VSA

Let $f : R \to R$ be defined as $f(x) = \frac{x^2 + 1}{2}$, then check whether $f$ is one-one or onto.

LA

Let $A = R - \{3\}$ and $B = R - \{1\}$. Let $f: A \to B$ defined as $f(x) = \frac{x-2}{x-3} \forall x \in A$. Then show that $f$ is bijective. [Outside India 2017]

SA

Prove that the function $f : R \to R$ given by $f(x) = x^3$ is bijective.

LA

Check whether the relation $R$ in the set $N$ of natural numbers given by $R = \{(a, b) : a \text{ is divisor of } b\}$ is reflexive, symmetric or transitive. Also determine whether $R$ is an equivalence relation. [Delhi 2020]

VSA

Let $R$ be the equivalence relation in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$. Write the equivalence class $[0]$. [All India 2018]

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