Question
5. Let 𝑅𝑅 be the equivalence relation on the set 𝐴𝐴 = {1, 2, 3, 4, 5, 6}, given by 𝑅𝑅 =
{(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 𝐴𝐴 induced by 𝑅𝑅 (i.e., find the quotient set 𝐴𝐴/𝑅𝑅).
Answers
GPT 3.5
The partition of A induced by R is the set of all equivalence classes of R.
To find the equivalence classes, we can start by considering each element in A one by one and see which elements it is related to.
1 is related to itself and 5, so the equivalence class [1] = {1, 5}.
2 is related to itself, 3, and 6, so the equivalence class [2] = {2, 3, 6}.
3 is related to itself, 2, and 6, so the equivalence class [3] = {2, 3, 6}.
4 is related to itself, so the equivalence class [4] = {4}.
5 is related to itself and 1, so the equivalence class [5] = {1, 5}.
6 is related to itself, 2, and 3, so the equivalence class [6] = {2, 3, 6}.
Therefore, the partition of A induced by R is {{1, 5}, {2, 3, 6}, {4}}.
To find the equivalence classes, we can start by considering each element in A one by one and see which elements it is related to.
1 is related to itself and 5, so the equivalence class [1] = {1, 5}.
2 is related to itself, 3, and 6, so the equivalence class [2] = {2, 3, 6}.
3 is related to itself, 2, and 6, so the equivalence class [3] = {2, 3, 6}.
4 is related to itself, so the equivalence class [4] = {4}.
5 is related to itself and 1, so the equivalence class [5] = {1, 5}.
6 is related to itself, 2, and 3, so the equivalence class [6] = {2, 3, 6}.
Therefore, the partition of A induced by R is {{1, 5}, {2, 3, 6}, {4}}.