VSA

A relation R on $A=\{1,2,3\}$ is defined as $R=\{(1,1),(3,3),(1,2)\}$. Is R a symmetric relation? Justify. Write the smallest relation set $R_{1}$ such that $R\cup R_{1}$ becomes an equivalence relation on the set {1, 2, 3}.

SA

Prove that the function $f$ is surjective, where $f : N \to N$ such that $f(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases}$.

LA

Let $A = R - \{5\}$ and $B = R - \{1\}$. Consider the function $f : A \to B$, defined by $f(x) = \frac{x - 3}{x - 5}$. Show that $f$ is one - one and onto. [All India 2024]

LA

Let $R$ be relation defined on the set of natural number $N$ as follows, $R = \{(x, y) : x \in N, y \in N, 2x + y = 41\}$. Find the domain and range of the relation $R$. Also verify whether $R$ is reflexive, symmetric and transitive. [Exemplar]

LA

Show that the function $f : R \to \{x \in R : -1 < x < 1\}$ defined by $f(x) = \frac{x}{1 + |x|}, x \in R$ is one-one and onto function.

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