Question

Let R is the equivalence relation in the set \[A = \left\{ {0,1,2,3,4,5} \right\}\] given by \[R = \left\{ {\left( {a,b} \right):2{\text{ divides }}\left( {a - b} \right)} \right\}\]. Write the equivalence class \[\left[ 0 \right]\].

Answer 1

Hint:
First, we will put \[b = 0\] in the given relation R to find the equivalence class \[\left[ 0 \right]\], Then we will find the multiples of 2 from the given set and then we will form a one set using the obtained multiples of 2.

Complete step by step solution:
We are given that R is the equivalence relation in the set \[A = \left\{ {0,1,2,3,4,5} \right\}\] given by \[R = \left\{ {\left( {a,b} \right):2{\text{ divides }}\left( {a - b} \right)} \right\}\].
To find the equivalence class \[\left[ 0 \right]\], putting \[b = 0\] in the given relation R, we get
So, we have \[a - 0\] is a multiple of 2.
This implies that \[a\] is a multiple of 2.
Now we will find the multiples of 2 from the given set, we get
0, 2, 4

Now taking them as one set to find the equivalence class \[\left[ 0 \right]\] is \[\left\{ {0,2,4} \right\}\].

Note:
The mathematical meaning of equivalence relation is a relation between elements of a set which is reflexive, symmetric, and transitive and which defines exclusive classes whose members bear the relation to each other and not to those in other classes. The key point is to put \[b = 0\] in the given relation R to find the equivalence class \[\left[ 0 \right]\],