Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
Is A ∪ B = B. because A ∪ B, Means A or B or Both ?
The rigorous way of showing that "$\displaystyle X= Y$" is to show that "$\displaystyle X\subseteq Y$" and that "$\displaystyle Y\subseteq X$". And to show that "$\displaystyle X\subseteq Y$" start with "if $\displaystyle x\in X$ and use the conditions on X and Y to conclude "therefore $\displaystyle x\in Y$".
So to show that, $\displaystyle A\cup B= B$ you must first show $\displaystyle A\cup B\subseteq B$. And you do that by saying "if $\displaystyle x\in A\cup B$ then either $\displaystyle x\in A$ or $\displaystyle x\in B$ (by definition of "$\displaystyle A\cup B$" and then do it in two cases:
1) If $\displaystyle x\in B$ we are done.
2) if $\displaystyle x\in A$ then because $\displaystyle A\subseteq B$, $\displaystyle x\in B$.
So that in either case, if $\displaystyle x\in A\cup B$ then $\displaystyle x\in B$.
All that remains is to show that $\displaystyle B\subseteq A\cup B$. To do that, if $\displaystyle x\in B$ then $\displaystyle x\in A\cup B$ by definition of $\displaystyle A\cup B$.