Question

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!

Answer 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.

Answer 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!

Answer 3

Thank you for the reassurance.

Answer 4

Keep up the good work!