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

1. ## 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 ?

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

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 $\displaystyle X$ is a set and $\displaystyle Y$ is any other set then $\displaystyle X\subseteq (X\cup Y)$.

So in the case $\displaystyle B\subseteq (A\cup B)$. Now all you need do is show $\displaystyle A\cup B\subseteq B.$

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

Originally Posted by Plato
Now all you need do is show $\displaystyle A\cup B\subseteq B.$
Is the answer because A U B means that A or B so it can be included in B ?

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

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

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

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 $\displaystyle A\cup B$.

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# show that if a and b are sets with a âŠ† b then

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