SA

Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y) \text{ is divisible by } n, x, y\in Z\}$. Determine if R is an equivalence relation.

LA

Show that $f:R\rightarrow R$ defined as $f(x)=\frac{x}{\sqrt{1+x^{2}}}$ is one-one but not onto.

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 $A=\mathbb{R}-\{3\}$ and $B=\mathbb{R}-\{1\}$. A function $f:A\rightarrow B$ is defined by $f(x)=(\frac{x-2}{x-3})$. Find whether f is one-one and onto.

LA

Let $L$ be the set of all lines in $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$. [NCERT]

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