Let X = {1, 2, 3, 4, 5, 6}. Which of the following could be an equivalence class of an equivalence relation on X? O {(1, 2), (3, 4), (5, 6)) O (1,3,5) O [1, 2), (3, 4), (5, 6}} O (12)(34)(56)

Question 2 List all possible equivalence relations on the set {1,2,3,4} without repetition up to isomorphism. Justify your answer.

) 7. Let A = {1,2,3,4}×{1,2,3,4}. Define an equivalence relation - by (x1,x2) ~ (X3,x4) iff x1x2 = xx4. List the elements of the equivalence class [(2,2)].

1. Let A = {1,2,3,4,5,6), and consider the following equivalence relation 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)} List the equivalence classes of R. 2. Let A = {a,bed el Suppose A is an equivalence relation on 4. Suppose R bes two equivaler.se alasses. Also wad, Re and aRd Write out R as a cot 3. Let A = {a,b,c,d,e}. Suppose R is an equivalence relation on A. Suppose R has three equivalence classes. Also aRd and bRc. Write out R as a set. 4. Let A {a,b,c,d,e). Suppose A is an equivalence relation on 4. Suppose also that anta and one, aRa and cite. How many equivalones classes dees At have! CIV 5. There are two different equivalence relations on the set A = {a,b}. Describe them. Diagrams will suffice.

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Let S be a nonempty subset of Z and let R be a relation defined on S by xRy if 3 | (x + 2y). If S = {−7, −6, −2, 0, 1, 4, 5, 7}, then what are the distinct equivalence classes in this case? Please show how you find them

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