Proving an Equivalence Relation: Tips & Examples

[helpm3pl3ase]

Iam not clear on how to prove a equivalence relation? I know that is has to have three properties reflexivity, symmetry, and transtivity, but I am unsure how to check.

For example Iam given f is a function from z to d. R(z,d) = binary relation

How do I prove that R(z,d) is a equivalence relation?? Iam unclear on how to approach this and work with the properties?

 

[d_leet]

A relation ~ on a set A, is an equivalence relation if

Let x,y,z belong to A

i) x~x (reflexive)
ii)If x~y then y~x (symmetric)
iii)if x~y and y~z then x~z (transitive)

To show that a given relation is an equivalence relation you would need to show that these 3 properties hold true for the relation.

 

[HallsofIvy]

As both you and d_leet said, you would prove R is an equivalence relation by showing that each of the conditions holds. How you would do that depends strongly on what R is!

"For example Iam given f is a function from z to d. R(z,d) = binary relation"
How is R(z, d) related to f? Do you mean R(z,d) if and only if d= f(z)?

Rather than saying "f is a function from z to d" it would be better to say "f is a function on set A" (with z and d members of set A).

If, indeed R(z,d) if and only if d= f(z), then you must prove:
1. Reflexive. That f(z)= z so R(z,z) for every member of set A.
2. Symmetric. If d= f(z), then z= f(d).
3. Reflexive. If x= f(z) and y= f(x), then y= f(z).

From 2, it looks like we are saying that f must be invertible. If that is the case, then for any d in A, there is only one x, such that d= f(x). Combining that with 1, we clearly must have f(x)= x for any member of A. That would mean that R(d,x) is equality: d= x.