Let A = {1, 2, 3, 4, 5, 6}, and consider the following equivalence relation R on A: R = { (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (2, 3), (3, 2), (4, 5), (5, 4), (4, 6), (6, 4), (5, 6), (6, 5) }. Show the partition of A defined by the equivalence classes of R.

A = C₁ U C₂ U C₄ =,

C₁ = 'class of one' = {1}

C₂ = 'class of two' = {2,3}

C₄ = 'class of four' = {4,5,6}

Step-by-step explanation:

We need to find the equivalence classes of A. Since 1 does only relates to itself, then the class of 1 only has 1 element, which is 1 itself.

2, apart from itself, only does relate with 3. Thus, the equivalence class of 2 contains 2 elements: 2 and 3 (note that 3 also relates only with 2 apart fromm itself.

The remaining 3 elements, 4, 5 and 6 relate between each other, thus, they form an entire equivalence class, the equivalence class of 4.

Those are the equivalence classes of A.