Question

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}.

Answer 1

We observe the following properties of R.

Reflexivity:

$$\begin{gathered} \mathrm{Let}a\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}R.Then, \\ a\in R \\ \Rightarrow \left(a,a\right)\in R\mathrm{for}\mathrm{all}a\in A \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}\mathrm{A}. \\ \mathrm{Symmetry}\mathit{:}\mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow \mathrm{Both}a\mathrm{and}b\mathrm{are}\mathrm{either}\mathrm{even}\mathrm{or}\mathrm{odd}. \\ \Rightarrow \mathrm{Both}b\mathrm{and}\mathit{}a\mathrm{are}\mathrm{either}\mathrm{even}\mathrm{or}\mathrm{odd}. \\ \Rightarrow \left(b,a\right)\in R\mathrm{for}\mathrm{all}a,b\in A \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}\left(a,b\right)\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow \mathrm{Both}a\mathrm{and}b\mathrm{are}\mathrm{either}\mathrm{even}\mathrm{or}\mathrm{odd}\mathrm{and}\mathrm{both}b\mathrm{and}c\mathrm{are}\mathrm{either}\mathrm{even}\mathrm{or}\mathrm{odd}. \\ \mathit{\Rightarrow }a,b\mathrm{and}c\mathrm{are}\mathrm{either}\mathrm{even}\mathrm{or}\mathrm{odd}. \\ \mathit{\Rightarrow }a\mathrm{and}c\mathrm{both}\mathrm{are}\mathrm{either}\mathrm{even}\mathrm{or}\mathrm{odd}. \\ \Rightarrow \left(a,c\right)\in R\mathrm{for}\mathrm{all}a,c\in A \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$

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.