Question

5. Let 𝑅 be the equivalence relation on the set 𝐴 = {1, 2, 3, 4, 5, 6}, given by 𝑅 =
{(1,1), (1,5), (2,2), (2,3), (2,6), (3,2), (3,3), (3,6), (4,4), (5,1), (5,5), (6,2), (6,3), (6,6)}.
Then, find the partition of 𝐴 induced by 𝑅 (i.e., find the quotient set 𝐴/𝑅).

Answer 1

The partition of A induced by R can be found by grouping together the elements that are equivalent under R.

First, let's identify the equivalence classes of R:

[1] = {1, 5}
[2] = {2, 3, 6}
[4] = {4}

We can see that the set {1, 5} forms an equivalence class because (1, 1) and (1, 5) are both in R.
Similarly, the set {2, 3, 6} forms an equivalence class because (2, 2), (2, 3), (2, 6), (3, 2), (3, 3), (3, 6), (6, 2), (6, 3), and (6, 6) are all in R.
Lastly, the set {4} forms an equivalence class because (4, 4) is in R.

Therefore, the partition of A induced by R is { {1, 5}, {2, 3, 6}, {4} }.