Question

prove that f of A union B is equals to f of A union f of B​

Answer 1

Answer:

Remember that A=B if and only if x∈A⟺x∈B, so to verify an equality of sets we need to see that the elements are on both sets, and to show that two sets are different we can just point out an element in one and not in the other.

To see that f(A∪B)=f(A)∪f(B), note that:

f(x)∈f(A∪B)⟺x∈A∪B⟺x∈A or x∈B⟺f(x)∈f(A) or f(x)∈f(B)⟺f(x)∈f(A)∪f(B)

On the other hand, if f(x)∈f(A)∩f(B) it just means that for some a∈A we have f(a)=f(x) and for some b∈B we have f(b)=f(x). We can use this to show that this is not the same set.

That is, we can find a function f:X→Y, and two sets A,B⊆X with some x∈A and x∉B, and y∈B for which y∉A such that f(x)=f(y). Now try to see why f(A∩B)≠f(A)∩f(B).

Answer 2

Answer:

what is union I don't know sorry sorry I