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Show that the relation $R$ in the $s e t=\{1,2,3,4,5\}$ given by $\mathbf{R}=\{(\mathbf{a}, \mathbf{b}):|\mathbf{a}-\mathbf{b}|$ is even $\}$, is an equivalance relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other. But no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$
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$\mathrm{A}=\{1,2,3,4,5\}$ and $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}):|\mathrm{a}-\mathrm{b}|$ is even $\}$
$\mathrm{R}=\{(1,3),(1,5),(3,5),(2,4)\}$
(a) (i) Let us take any element of a set A.
then $|a-a|=0$ which is even.
$\Rightarrow R$ is reflexive.
(ii) If $|\mathrm{a}-\mathrm{b}|$ is even, then $|\mathrm{b}-\mathrm{a}|$ is also even, where, $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}):|\mathrm{a}-\mathrm{b}|$ is even $\} \Rightarrow \mathrm{R}$ is symmetric.
(iii) Further $a-c=a-b+b-c$
If $|\mathrm{a}-\mathrm{b}|$ and $|\mathrm{b}-\mathrm{c}|$ are even, then their sum $|a-b+b-c|$ is also even.
$\Rightarrow|\mathrm{a}-\mathrm{c}|$ is even, $\quad \therefore \mathrm{R}$ is transitive.
Hence $R$ is an equivalence relation.
(b) Elements of $\{1,3,5\}$ are related to each other.
Since $|1-3|=2,|3-5|=2,|1-5|=4$. All are even numbers.
$=$ Elements of $\{1,3,5\}$ are related to each other. Similarly elements of $\{2,4\}$ are related to each other. Since $|2-4|$ $=2$ an even number. No element of set $\{1,3,5\}$ is related to any element of $\{2,4\}$.
$\mathrm{R}=\{(1,3),(1,5),(3,5),(2,4)\}$
(a) (i) Let us take any element of a set A.
then $|a-a|=0$ which is even.
$\Rightarrow R$ is reflexive.
(ii) If $|\mathrm{a}-\mathrm{b}|$ is even, then $|\mathrm{b}-\mathrm{a}|$ is also even, where, $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}):|\mathrm{a}-\mathrm{b}|$ is even $\} \Rightarrow \mathrm{R}$ is symmetric.
(iii) Further $a-c=a-b+b-c$
If $|\mathrm{a}-\mathrm{b}|$ and $|\mathrm{b}-\mathrm{c}|$ are even, then their sum $|a-b+b-c|$ is also even.
$\Rightarrow|\mathrm{a}-\mathrm{c}|$ is even, $\quad \therefore \mathrm{R}$ is transitive.
Hence $R$ is an equivalence relation.
(b) Elements of $\{1,3,5\}$ are related to each other.
Since $|1-3|=2,|3-5|=2,|1-5|=4$. All are even numbers.
$=$ Elements of $\{1,3,5\}$ are related to each other. Similarly elements of $\{2,4\}$ are related to each other. Since $|2-4|$ $=2$ an even number. No element of set $\{1,3,5\}$ is related to any element of $\{2,4\}$.
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