Discrete Math

Which of these relations on {0, 1, 2, 3} are equivalence relations? Justify the relation(s) that are not equivalent.

R1: {(0,0), (1,1), (2,2), (3,3)}
R2: {(0,0), (1,1), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3)}
R3: {(0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,2), (3,3)}

R1: This relations is equivalent
R2: This relation is equivalent
R3: This relation is not equivalent because:
• It is reflexive because the relation does contains (0,0), (1,1), (2,2), and (3,3).
• It is not symmetric because the relation contains (1,2), but not (2,1).
•This relation is transitive.

I think something is not right. . .Any suggestions? Thanks for any helpful replies!

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  1. I think I may have found the problem in my thinking:

    R2 is not equivalent right? Because it is not transitive.

    Justification:
    It is reflexive because the relation does contain (0,0), (1,1), (2,2), and (3,3).
    It is symmetric because the relation contains (1,3) ⋏ (3,1), and (2,3) ⋏ (3,2)
    Though the relation contains (1,3) ⋏ (3,2) it does not have (1,2), which means it is not transitive.

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  2. R1: This relations is equivalent (agree)

    R2 is not equivalent right? Because it is not transitive. (agree)

    R3: This relation is not equivalent because the relation contains (1,2), but not (2,1) (agree)

    Excellent!

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  3. Thank you for the reassurance.

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  4. Keep up the good work!

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