# Show that if A and B are sets with A ⊆ B, then A ∪ B = B.

• October 5th 2013, 12:43 PM
lamentofking
Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
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 ?
• October 5th 2013, 01:03 PM
Plato
Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
Quote:

Originally Posted by lamentofking
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 ?

Basically, yes.
Realize, That if $X$ is a set and $Y$ is any other set then $X\subseteq (X\cup Y)$.

So in the case $B\subseteq (A\cup B)$. Now all you need do is show $A\cup B\subseteq B.$
• October 6th 2013, 01:45 PM
lamentofking
Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
Quote:

Originally Posted by Plato
Now all you need do is show $A\cup B\subseteq B.$

Is the answer because A U B means that A or B so it can be included in B ?
• October 6th 2013, 01:55 PM
Plato
Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
Quote:

Originally Posted by lamentofking
Is the answer because A U B means that A or B so it can be included in B ?

Yes.
• October 6th 2013, 04:48 PM
HallsofIvy
Re: Show that if A and B are sets with A ⊆ B, then A ∪ B = B.
The rigorous way of showing that " $X= Y$" is to show that " $X\subseteq Y$" and that " $Y\subseteq X$". And to show that " $X\subseteq Y$" start with "if $x\in X$ and use the conditions on X and Y to conclude "therefore $x\in Y$".

So to show that, $A\cup B= B$ you must first show $A\cup B\subseteq B$. And you do that by saying "if $x\in A\cup B$ then either $x\in A$ or $x\in B$ (by definition of " $A\cup B$" and then do it in two cases:
1) If $x\in B$ we are done.

2) if $x\in A$ then because $A\subseteq B$, $x\in B$.

So that in either case, if $x\in A\cup B$ then $x\in B$.

All that remains is to show that $B\subseteq A\cup B$. To do that, if $x\in B$ then $x\in A\cup B$ by definition of $A\cup B$.