# Union and Intersection of Sets

Lesson

Having just looked at the notation and language we use to describe sets, we now move on to think about what happens when we combine sets together.  Now this isn't as strange as you may first think.

Imagine you have the following clothes in your wardrobe. This is our universal set.

From this we could create two sets, the set $P=\left\{\text{shirts in your wardrobe}\right\}$P={shirts in your wardrobe}

and the set $Q=\left\{\text{blue clothes in your wardrobe}\right\}$Q={blue clothes in your wardrobe}

We can also display these in a Venn Diagram. Imagine you are sorting them into two groups.

So we begin sorting our clothes,

But where do I put the blue shirt? In the shirts set? or in the Blue clothes set?  The answer is to put it in the overlapping piece.  This is the piece that displays what belongs in both sets.

This is a visual explanation of the idea of the INTERSECTIONS of sets.

## Intersections

Just like how a road intersection is the place where two roads cross paths, an intersection of sets is where two sets overlap. Elements that appear in the intersection of sets are elements that have the same characteristic as both the individual sets, just like our blue shirt from above.

Mathematically we write the intersection of sets using the intersection symbol, $\cap$.  We interpret the intersection of $A$A and $B$B, $A\cap B$AB to be what appears in both set $A$A and set $B$B. It helps some students to relate $\cap$ to AND or to think of the symbol like a bridge joining both sets.

We can also consider the intersection of numerical sets.

##### Example 1

$A=\left\{5,10,15,20,25,30\right\}$A={5,10,15,20,25,30} and $B=\left\{6,12,18,24,30\right\}$B={6,12,18,24,30} then $A\cap B=\left\{30\right\}$AB={30}

Incidently this is a set notation version of finding the lowest common multiple between two numbers.

##### Example 2

$M=\left\{\text{even numbers up to 30}\right\}$M={even numbers up to 30}, $N=\left\{\text{factors of 24}\right\}$N={factors of 24}, then $M\cap N=\left\{\text{the even factors of 24}\right\}$MN={the even factors of 24}

We could write out all the members if we wanted...

$M=\left\{2,4,6,8,10,12,14,16,18,20,22,24,26,28,30\right\}$M={2,4,6,8,10,12,14,16,18,20,22,24,26,28,30}, $N=\left\{1,2,3,4,6,8,12,24\right\}$N={1,2,3,4,6,8,12,24} and $M\cap N=\left\{2,4,6,8,12,24\right\}$MN={2,4,6,8,12,24}

##### Example 3

$P=\left\{\text{even integers}\right\}$P={even integers}, $Q=\left\{\text{negative integers}\right\}$Q={negative integers}, then $P\cap Q=\left\{\text{even negative integers}\right\}$PQ={even negative integers}

In this case we cannot write the sets out in full as they are all infinite sets.

## Unions

If we consider the intersection the AND of mathematical sets, then the union is the OR.  $A\cup B$AB is the notation we use, and we would read this as either the union of $A$A and $B$B or, the elements in $A$A or $B$B .

$A\cup B$AB is the set of the elements that are in either $A$A or $B$B.

With regards to our wardrobe example above, the union of the set of shirts and blue clothes is the set of all clothes in my wardrobe that is either blue or a shirt.

##### Example 4

$A=\left\{5,11,16,17,20,25\right\}$A={5,11,16,17,20,25} and $B=\left\{4,12,15,25,30\right\}$B={4,12,15,25,30} then $A\cup B=4,5,11,12,15,16,17,20,25,30$AB=4,5,11,12,15,16,17,20,25,30

##### Example 5

$M=\left\{\text{even integers}\right\}$M={even integers} and $N=\left\{\text{odd integers}\right\}$N={odd integers} then $M\cup N=\left\{\text{integers}\right\}$MN={integers}

#### Worked Examples

##### question 1

The following questions relate to unions and intersections of sets.

1. What is meant by the intersection of sets $A$A and $B$B?

The set of elements that are members of set A or set B or both sets.

A

The set of elements that are common to set A and set B.

B

The set of elements that are only members of one of set A or set B.

C

The set of elements that are not members of either set A or set B.

D
2. Suppose set $A$A$=$=$\left\{3,4,5,6,7\right\}${3,4,5,6,7} and set $B$B$=$=$\left\{3,7,8,9\right\}${3,7,8,9}

Which answer correctly shows the intersection of the two sets?

$\left\{3,4,5,6,7\right\}\cup\left\{3,7,8,9\right\}${3,4,5,6,7}{3,7,8,9}$=$=$\left\{3,7\right\}${3,7}

A

$\left\{3,4,5,6,7\right\}\cap\left\{3,7,8,9\right\}${3,4,5,6,7}{3,7,8,9}$=$=$\left\{3,4,5,6,7,8,9\right\}${3,4,5,6,7,8,9}

B

$\left\{3,4,5,6,7\right\}\cup\left\{3,7,8,9\right\}${3,4,5,6,7}{3,7,8,9}$=$=$\left\{3,4,5,6,7,8,9\right\}${3,4,5,6,7,8,9}

C

$\left\{3,4,5,6,7\right\}\cap\left\{3,7,8,9\right\}${3,4,5,6,7}{3,7,8,9}$=$=$\left\{3,7\right\}${3,7}

D

##### question 2

If $A$A is the set of factors of $12$12, and $B$B is the set of factors of $18$18, then list the elements of:

1. $B\cup A$BA

2. $A\cap B$AB

##### question 3

Let $A$A be the set of squares less than $70$70, and $B$B be the set of integers less than $20$20.

1. Which set is $\left(A\cap B\right)'$(AB)?

The empty set.

A

The set of squares higher than $70$70.

B

The set of all integers except for $1$1, $4$4, $9$9 and $16$16.

C
2. Which set is $A'\cap B'$AB?

The set of square numbers greater than or equal to $70$70.

A

The set of integers that are greater than or equal to $20$20 and are not squares less than $70$70.

B

The set of integers that are greater than $70$70.

C

The empty set.

D
3. Is $A'\cap B'$AB the same as $\left(A\cap B\right)'$(AB)?

Yes

A

No

B