LA

Show that a function $f : N \to N$ be defined by $f(n) = \begin{cases} 2n, & n = 2, 4, 6, 8, \dots \\ n - 1, & n = 3, 7, 11, 15, \dots \\ \frac{n + 1}{2}, & n = 1, 5, 9, 13, \dots \end{cases}$ then $f$ is bijective. [Competency]

SA

If $f:R^{+}\rightarrow R$ is defined as $f(x) = \log_{a} x$ ($a > 0$ and $a\ne1$), prove that f is a bijection. ($R^{+}$ is a set of all positive real numbers.)

VSA

Check whether $f : Z \times Z \to Z \times Z$ (where $Z$ is the set of integers) defined as $f(x, y) = (2y, 3x)$ is injective or not. [Delhi 2026]

SA

Let R be a relation defined over N, where N is set of natural numbers, defined as "mRn if and only if m is a multiple of n, m, $n\in N$." Find whether R is reflexive, symmetric and transitive or not.

VSA

Check whether $f : R - \{3\} \to R$ defined as $f(x) = \frac{x - 2}{x - 3}$ is onto or not. [Delhi 2026]

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