Answer:
a is not an equivalence relation.
b is an equivalence relation.
Step-by-step explanation:
a.
A = {1,2,3,4, 5) R={(1,1), (1,2), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3), (4,4), (5,5)
To see if is an equivalence relation you need to see if you have these 3 things:
Part 1: xRx for all x in A. This is the reflexive property.
Do we? Yes we have all these points in R: (1,1), (2,2) ,(3,3) ,(4,4), and (5,5).
Part 2: If xRy then yRx. This is the symmetic property.
Do we? We have (1,2) but not (2,1). So it isn't symmetric.
Part 3: If xRy and yRz then xRz.
Do we? We are not going to check this because there is no point. We have to have all 3 parts fot it be an equivalence relation.
b.
A = {a, b, c} R={(a, a), (a, c), (b, b), (c, a), (c, c)}
To see if is an equivalence relation you need to see if you have these 3 things:
Part 1: xRx for all x in A. This is the reflexive property.
Do we? Yes we have all these points in R: (a,a),(b,b), and (c,c).
Part 2: If xRy then yRx. This is the symmetric property.
Do we? We have (a,c) and (c,a). We don't need to worry about any other (x,y) since there are no more with x and y being different. This is symmetric.
Part 3: If xRy and yRz then xRz.
Do we? We do have (a,c), (c,a), and (a,a).
We do have (c,a), (a,c), and (c,c).
So it is transitive.
Question b has all 3 parts so it is an equivalence relation.
