Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.
We observe the following properties of R.
Reflexivity:
Let $a$ be an arbitrary element of $R$. Then,\
$a \in R$
$\Rightarrow(a, a) \in R$ for all $a \in A$
So, $R$ is reflexive on $\mathrm{A}$.
Symmetry : Let $(a, b) \in R$
$\Rightarrow$ Both $a$ and $b$ are either even or odd.
$\Rightarrow$ Both $b$ and $a$ are either even or odd.
$\Rightarrow(b, a) \in R$ for all $a, b \in A$
So, $R$ is symmetric on $A$.
Transitivity: Let $(a, b)$ and $(b, c) \in R$
$\Rightarrow$ Both $a$ and $b$ are either even or odd and both $b$ and $c$ are either even or odd.
$\Rightarrow a, b$ and $c$ are either even or odd.
$\Rightarrow a$ and $c$ both are either even or odd.
$\Rightarrow(a, c) \in R$ for all $a, c \in A$
So, $R$ is transitive on $A$.
Thus, R is an equivalence relation on A.
We observe that all the elements of the subset {1, 3, 5, 7} are odd. Thus, they are related to each other.
This is because the relation R on A is an equivalence relation.
Similarly, the elements of the subset {2, 4, 6} are even. Thus, they are related to each other because every element is even.
Hence proved.