Question
5. Let 𝑅𝑅 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).
{(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).
Answers
GPT 3.5
The partition of A induced by R is determined by the equivalence classes of R.
An equivalence class of R is a subset of A that contains all the elements that are related to each other by R.
So, let's find the equivalence classes of R:
- The element 1 is related to itself and to the element 5. So, the equivalence class of 1 is {1, 5}.
- The element 2 is related to itself, to the element 3, and to the element 6. So, the equivalence class of 2 is {2, 3, 6}.
- The element 3 is related to itself, to the element 2, and to the element 6. So, the equivalence class of 3 is also {2, 3, 6}.
- The element 4 is related to itself. So, the equivalence class of 4 is {4}.
- The element 5 is related to itself. So, the equivalence class of 5 is {5}.
- The element 6 is related to itself, to the element 2, and to the element 3. So, the equivalence class of 6 is also {2, 3, 6}.
The partition of A induced by R is then: {{1, 5}, {2, 3, 6}, {4}}.
An equivalence class of R is a subset of A that contains all the elements that are related to each other by R.
So, let's find the equivalence classes of R:
- The element 1 is related to itself and to the element 5. So, the equivalence class of 1 is {1, 5}.
- The element 2 is related to itself, to the element 3, and to the element 6. So, the equivalence class of 2 is {2, 3, 6}.
- The element 3 is related to itself, to the element 2, and to the element 6. So, the equivalence class of 3 is also {2, 3, 6}.
- The element 4 is related to itself. So, the equivalence class of 4 is {4}.
- The element 5 is related to itself. So, the equivalence class of 5 is {5}.
- The element 6 is related to itself, to the element 2, and to the element 3. So, the equivalence class of 6 is also {2, 3, 6}.
The partition of A induced by R is then: {{1, 5}, {2, 3, 6}, {4}}.