A quotient set, also known as a factor set or partition set, is a set obtained by partitioning another set into equivalence classes.
Let Relation R be on the Set of Integers.
R={(x,y)∣x≡ymod5}.
The Relation contains ordered pairs of integers where the first integer (x) is congruent to the second integer (y) modulo 5, meaning that both leave the same remainder when divided by 5.
To find the quotient set, we need to partition the set into equivalence classes based on the relation.In this case, each equivalence class consists of elements that are congruent to each other modulo 5.
So, the equivalence classes will be:
- The set of integers congruent to 0 modulo 5: [0]={...,−10,−5,0,5,10,...}
- The set of integers congruent to 1 modulo 5: [1]={...,−9,−4,1,6,11,...}
- The set of integers congruent to 2 modulo 5: [2]={...,−8,−3,2,7,12,...}
- The set of integers congruent to 3 modulo 5: [3]={...,−7,−2,3,8,13,...}
- The set of integers congruent to 4 modulo 5: [4]={...,−6,−1,4,9,14,...}
So, the quotient set in set-builder form will be: {[0],[1],[2],[3],[4]}
