SA

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}$. If the relation $R$ is an equivalence relation then what can be said about $\alpha$.

LA

Let $N$ be the set of all natural numbers and $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d) \Leftrightarrow ad = bc$ for all $(a, b), (c, d) \in N \times N$. Show that $R$ is an equivalence relation on $N \times N$. Also, find the equivalence class of $(2, 6)$, i.e., $[(2, 6)]$. [All India 2019, 2022]

LA

Show that a function $f : R \to R$ defined by $f(x) = \frac{2x}{1 + x^2}$ is neither one-one nor onto. Further, find set $A$ so that the given function $f : R \to A$ becomes an onto function. [All India 2023, Delhi 2024]

SA

Let $R$ be a relation defined over $N$, where $N$ is set of natural numbers, defined as "$m R n$ if and only if $m$ is a multiple of $n$, where $m, n \in N$." Find whether $R$ is reflexive, symmetric and transitive or not. [Delhi 2025]

VSA

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

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