The "standard" way to prove set X= set Y is to show "X is a subset of Y" and "Y is a subset of X". And the "standard" way to show "X is a subset of Y" is to start "if x is in set X" and use the definitions of X and Y to conclude "x is in Y".

So: If then and [tex]x\in B\cup D/tex]. Since , either or .

case 1) . Since and , .

case 2) . Since and , .

In either case, so .

Now, if then either or .

case 1) . Then and . Since , . Since [tex]x\in A[tex] and , .

case 2) . Then and . Since , . Since [tex]x\in A[tex] and , .