Question

if a and b are subsets what is a union b​

Answer 1

Look closely at your argument: it implies that x∈A∪B if and only if x∈A, which is clearly not the case in general. For instance, if A={0} and B={1}, then 1∈A∪B, but 1∉A.

You’re starting in the wrong place. To show that X⊆Y, you need to show that every member of X is a member of Y. To do this you don’t start with an arbitrary member of Y: some of them may not be in X. You start with an arbitrary member of X and try to show that it must belong to Y. Here you want to show that A⊆A∪B, so you should be starting with an arbitrary member of A. And when you do that, the rest of the argument is trivial: if x∈A, then it’s certainly true that x∈A or x∈B, which is what it means to say that x∈A∪B.

If you insist on doing the argument with formal logical expressions, what you want is

x∈A→(x∈A∨x∈B)↔x∈A∪B.

Answer 2

A union B

Suppose a=1,2

B=1,2

Then a union b

{1,2}{1,2}

1,2=a union b