LA

Show that the relation $S$ on the set $A = \{x \in Z : 0 \le x \le 12\}$ given by $S = \{(a, b) : a, b \in Z, |a - b| \text{ is divisible by } 3\}$ is an equivalence relation. [All India 2019]

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]

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.)

LA

Show that the function $f : N \to N$ given by $f(n) = n - (-1)^n$ for all $n \in N$ is a bijection. [Competency]

VSA

Let $A = \{1, 2, 3\}$. We define $R_1 = \{(1, 2), (3, 2), (1, 3)\}, R_2 = \{(1, 3), (3, 6), (2, 1), (1, 2)\}$. Then check which is/are the relation on $A$

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