Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence re - lation that the others lack. a) { (0,0), (1,1), (2,2), (3,3) } b) { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) } c) { (0,0), (1,1), (1,2), (2,1), (2,2), (3,3) } d) { (0,0), (1,1), (1,3), (2,2), (2,3), (3,1), (3,2), (3, 3) } e) { (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3) }

The relations that are equivalence relations are a) and c)

Step-by-step explanation:

A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive

We are going to analyze each one.

a) { (0,0), (1,1), (2,2), (3,3) }

Is an equivalence relation because it has all the properties.

b) { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) }

Is not an equivalence relation. Not reflexive: (1,1) is missing, not transitive: (0,2) and (2,3) are in the relation, but not (0,3)

c) { (0,0), (1,1), (1,2), (2,1), (2,2), (3,3) }

Is an equivalence relation because it has all the properties.

d) { (0,0), (1,1), (1,3), (2,2), (2,3), (3,1), (3,2) (3,3) }

Is not an equivalence relation. Not transitive: (1,3) and (3,2) are in the relation, but not (1,2)

e) { (0,0), (0,1) (0,2), (1,0), (1,1), (1,2), (2,0), (2,2), (3,3) }

Is not an equivalence relation. Not symmetric: (1,2) is present, but not (2,1) Not transitive: (2,0) and (0,1) are in the relation, but not (2,1)