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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
MCQ
The relation $R = \{(1, 1), (2, 2), (3, 3)\}$ on the set $\{1, 2, 3\}$ is:
  • A
    Symmetric only.
  • B
    Reflexive only.
  • ✓
    An equivalence relation.
  • D
    Transitive only.
✓

Answer

Correct option: C.
An equivalence relation.
$R = \{(a, b): a = b$ and $a, b \in\text{A}\}$
Reflexivity: Let $\text{a}\in\text{A}$
Here,
$a = a$
$\Rightarrow\ (\text{a, a})\in\text{R}$ for all $\text{a}\in\text{A}$
So$, R$ is reflexive on $A.$
Symmetry: Let $\text{a, b}\in\text{A}$ such that $ (\text{a, b})\in\text{R}.$ Then,
$ (\text{a, b})\in\text{R}$
$\Rightarrow\ \text{a}=\text{b}$
$\Rightarrow\ \text{b}=\text{a}$
$\Rightarrow\ (\text{b, a})\in\text{R}$ for all $\text{a}\in\text{A}$
So$, R$ is symmetric on $A.$​​​​​​​
Transitive: Let $\text{a, b, c}\in\text{A}$ such that $ (\text{a, b})\in\text{R}$ and $ (\text{b, c})\in\text{R}.$ Then,
$ (\text{a, b})\in\text{R}\Rightarrow\ \text{a}=\text{b}$
and $ (\text{b, c})\in\text{R}\Rightarrow\ \text{b}=\text{c}$
$\Rightarrow\ \text{a}=\text{c}$
$\Rightarrow\ (\text{a, c})\in\text{R}$ for all $\text{a}\in\text{A}$
So$, R$ is transitive on $A.$
Hence$, R$ is an equivalence relation on $A.$

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