SA

Let $A = R - \{2\}$ and $B = R - \{1\}$, consider the function $f : A \to B$ defined by $f(x) = \frac{x - 1}{x - 2}$. Is $f$ one-one and onto? [All India 2019]

LA

Check whether the relation S in the set of real numbers R defined by $S=\{(a,b)$: where $a-b+\sqrt{2}$ is an irrational number is reflexive, symmetric or transitive.

SA

Prove that the function $f : R \to R$ given by $f(x) = x^3$ is bijective.

LA

If $N$ denotes the set of all natural numbers and $R$ is the relation on $N \times N$ defined by $(a, b) R (c, d)$ if $ad(b + c) = bc(a + d)$. Show that $R$ is an equivalence relation. [Delhi 2015, All India 2023]

VSA

For the set $A = \{1, 2, 3\}$, define a relation $R$ in the set $A$ as follows: $R = \{(1, 1), (2, 2), (3, 3), (1, 3)\}$. Write the ordered pairs to be added to $R$ to make it the smallest equivalence relation.

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