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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
MCQ
The relation $'R\ '$ in $N \times N$ such that $(a, b)R(c, d) \Leftrightarrow a + d = b + c$ is:
  • A
    Reflexive but not symmetric.
  • B
    Reflexive and transitive but not symmetric.
  • ✓
    An equivalence relation.
  • D
    None of the these.
✓

Answer

Correct option: C.
An equivalence relation.
We observe the following properties of relation $R.$
Reflexivity: Let $(\text{a, b})\in\text{N}\times\text{N}$
$\Rightarrow\ \text{a, b}\in\text{N}$
$\Rightarrow\ \text{a}+\text{b}=\text{b}+\text{a}$
$\Rightarrow\ (\text{a, b})\in\text{R}$
So$, R$ is reflexive on $N \times N.$
Symmetry: Let $(\text{a, b}),\ (\text{c, d})\in\text{N}\times\text{N}$ such that $(a, b)R(c, d)$
$\Rightarrow\ \text{a}+\text{d}=\text{b}+\text{c}$
$\Rightarrow\ \text{d}+\text{a}=\text{c}+\text{b}$
$\Rightarrow\ (\text{d, c}),\ (\text{b, a})\in\text{R}$
So$, R$ is symmetric on $N \times N.$​​​​​​​
Transitivity: Let $(\text{a, b}),\ (\text{c, d}),\ (\text{e, f})\in\text{N}\times\text{N}$ such that $(a, b)R(c, d)$ and $(c, d)R(e, f)$
$\Rightarrow a + d = b + c$ and $c + f = d + e$
$\Rightarrow a + d + c + f = b + c + d + e$
$\Rightarrow a + f = b + e$
$\Rightarrow (a, b)R(e, f)$
So$, R$ is transitive on $N \times N.$
Hence$, R$ is an equivalence relation on $N.​​​​​​​$

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