Answer:
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)
where s is the side.