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# Set Theory: Union of Sets, Definition and Examples

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The union of two sets $A$ and $B$ is the set of elements that are either in $A$ or in $B$ or in both $A$ and $B$.

In other words, the unions of two sets $A$ and $B$ is the set of elements that are in at least one of $A$ and $B$.

The notation for the union of two sets $A$ and $B$ is $A\cup B$ or $A\bigcup B$ (just a bigger $\cup$ symbol). In set notation, we have:

$$A\cup B=\{x\mid x\in A \;{\rm or}\;x\in B\}$$

Which of the following is FALSE?

A

The set of all odd integers equals the union

$\{x\in{\mathbb Z}\mid\,x=4k+1, \;{\rm for\;some\;integer}\;k\}\cup\{x\in{\mathbb Z}\mid\,x=4k+3, \;{\rm for\;some\;integer}\;k\}$

B

The set of all odd integers is a subset of the union

$\{x\in{\mathbb Z}\mid\,x=2k+1, \;{\rm for\;some\;integer}\;k\}\cup\{x\in{\mathbb Z}\mid\,x=4k+2, \;{\rm for\;some\;integer}\;k\}$

C

The set of all even integers equals the union

$\{x\in{\mathbb Z}\mid\,x=4k+3=4\ell+1, \;{\rm for\;some\;integers}\;k,\ell\}\cup\{x\in{\mathbb Z}\mid\,x=2k, \;{\rm for\;some\;integer}\;k\}$

D

The set $S$ of titles of the albums by Sugababes. The union of the set $S_1$ of all movies starring Morgan Freeman and the set $S_2$ of all movies starring both Morgan Freeman and Brad Pitt equals the set $S_2=S_1\cup S_2$ of all movies starring both Morgan Freeman and Brad Pitt.

E

The union of the set $S_1$ of all movies starring Morgan Freeman and the set $S_2$ of all movies starring Brad Pitt equals the set $S=S_1\cup S_2$ of all movies starring either Morgan Freeman or Brad Pitt or both Morgan Freeman and Brad Pitt.