LA

A relation R on set $A=\{-4,-3,-2,-1,0,1,2,3,4\}$ be defined as $R=\{(x,y):x+y$ is an integer divisible by 2). Show that R is an equivalence relation. Also, write the equivalence class [2].

LA

A relation $R$ is defined on a set of real numbers $\mathbb{R}$ as $R = \{(x, y) : x \cdot y \text{ is an irrational number}\}$. Check whether $R$ is reflexive, symmetric and transitive or not. [All India 2024]

SA

Let $A = \{-1, 0, 1, 2\}$, $B = \{-4, -2, 0, 2\}$ and $f, g : A \to B$ be functions defined by $f(x) = x^2 - x, x \in A$ and $g(x) = 2 \left| x - \frac{1}{2} \right| - 1, x \in A$. Are $f$ and $g$ equal? Justify your answer.

VSA

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

LA

Let $f : \mathbb{R} - \left\{ \frac{4}{3} \right\} \to \mathbb{R}$ be a function defined as:$$f(x) = \frac{4x}{3x+4}$$Show that $f$ is a one-one function. Also, check whether $f$ is an onto function or not.

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