LA

Let $A = \{1, 2, 3, \dots, 9\}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$ for $(a, b), (c, d)$ in $A \times A$. Prove that $R$ is an equivalence relation. Also obtain the equivalence class $[(2, 5)]$. [Delhi 2014]

LA

A relation $R$ is defined on $N \times N$ (where $N$ is the set of natural numbers) as $(a, b) R (c, d) \Leftrightarrow a - c = b - d$. Show that $R$ is an equivalence relation. [Delhi 2024]

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 $N$ 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

Prove that the function $f : N \to N$, defined by $f(x) = x^2 + x + 1$ is one-one but not onto. [Delhi 2019]

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