Let R be a relation of X to Y

[Boris22]

Let R be a relation of X to Y
a) Show that R [A ∪ B] = R[A] ∪ R[ B ] if A, B ⊆ X.
b) If ∆A = {(a, a) | a ∈ A} (that is, ∆A is the relation in which each element a ∈ A is only related
with himself). Show that R ◦ ∆X = R = ∆Y◦ R

 

[Steven G]

You will receive help from the forum if you post what you have done so far. In that way we can see the method you want to use and see where you need help. Please back following the forum's guideline which you should have read.

 

[Boris22]

For a) it would be first to check that the domain and codomain are the same, and then verify that the correspondence rule is the same to ensure that the two parts of the equality are equivalent. The problem here is that I don't know how to give formal proof of this.

 

[pka]

Let R be a relation of X to Y
a) Show that R [A ∪ B] = R[A] ∪ R[ B ] if A, B ⊆ X.
b) If ∆A = {(a, a) | a ∈ A} (that is, ∆A is the relation in which each element a ∈ A is only related
with himself). Show that R ◦ ∆X = R = ∆Y◦ R

I would happily help if only you had told us what \(R[A]\) means.
You need to understand that notation is not standard, therefore you must define yours terms.