VSA

Find the maximum number of equivalence relations on the set $A = \{1, 2, 3\}$.

SA

Let $A = \{1, 2, 3\}$. Then, find the number of relations containing $(1, 2)$ and $(1, 3)$, which are reflexive and symmetric but not transitive.

LA

Let $A = R - \{5\}$ and $B = R - \{1\}$. Consider the function $f : A \to B$, defined by $f(x) = \frac{x - 3}{x - 5}$. Show that $f$ is one - one and onto. [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.

SA

Let $n$ be a fixed positive integer. Define a relation $R$ in $Z$ as follows $\forall a, b \in Z$, $a R b$ if and only if $a - b$ is divisible by $n$. Show that $R$ is an equivalence relation.

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