Search any question & find its solution
Question:
Answered & Verified by Expert
If we define a relation $\mathrm{R}$ on the set $\mathrm{N} \times \mathrm{N}$ as $(a, b) R(c, d) \Leftrightarrow a+d=b+c$ for $a l l(a, b),(c, d) \in N \times N$, then
the relation is:
Options:
the relation is:
Solution:
2907 Upvotes
Verified Answer
The correct answer is:
equivalence relation
$(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d}) \Leftrightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}$
(i) $\mathrm{a}+\mathrm{a}=\mathrm{a}+\mathrm{a}$.
$(\mathrm{a}, \mathrm{a}) \mathrm{R}(\mathrm{a}, \mathrm{a}) \Rightarrow \mathrm{R}$ is reflexive
(ii) $(a, b) R(c, d) \Rightarrow a+d=b+c$
$(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow \mathrm{c}+\mathrm{b}=\mathrm{d}+\mathrm{a}$
$\mathrm{R}$ is symmetric.
(iii) Let $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ and $(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{e}, \mathrm{f})$
$\Rightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}$ and $\mathrm{c}+\mathrm{f}=\mathrm{d}+\mathrm{e}$
$\Rightarrow a+d+c+f=b+c+d+e$
$\Rightarrow \mathrm{a}+\mathrm{f}=\mathrm{b}+\mathrm{e}$
$\Rightarrow(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{e}, \mathrm{f})$
R is transitive.
from (i), (ii), (iii) R is anequivalence relation.
(i) $\mathrm{a}+\mathrm{a}=\mathrm{a}+\mathrm{a}$.
$(\mathrm{a}, \mathrm{a}) \mathrm{R}(\mathrm{a}, \mathrm{a}) \Rightarrow \mathrm{R}$ is reflexive
(ii) $(a, b) R(c, d) \Rightarrow a+d=b+c$
$(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow \mathrm{c}+\mathrm{b}=\mathrm{d}+\mathrm{a}$
$\mathrm{R}$ is symmetric.
(iii) Let $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ and $(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{e}, \mathrm{f})$
$\Rightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}$ and $\mathrm{c}+\mathrm{f}=\mathrm{d}+\mathrm{e}$
$\Rightarrow a+d+c+f=b+c+d+e$
$\Rightarrow \mathrm{a}+\mathrm{f}=\mathrm{b}+\mathrm{e}$
$\Rightarrow(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{e}, \mathrm{f})$
R is transitive.
from (i), (ii), (iii) R is anequivalence relation.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.