wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
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}.

Open in App
Solution

We observe the following properties of R.

Reflexivity:

Let a be an arbitrary element of R. Then,aRa, aR for all aASo, R is reflexive on A.Symmetry: Let a, bRBoth a and b are either even or odd.Both b and a are either even or odd.b, aR for all a, bASo, R is symmetric on A.Transitivity: Let a, b and b, cRBoth a and b are either even or odd and both b and c are either even or odd.a, b and c are either even or odd.a and c both are either even or odd.a, c R for all a, cASo, 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.

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Definition of Function
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon