3 The intersection and union of sets

3.1 Intersection

Given two sets, $A$ and $B$ , the intersection of $A$ and $B$ is a set which contains elements that are common both to $A$ and $B$ . We write $A\cap B$ to denote the intersection of $A$ and $B$ . Mathematically we write this as:

Note that $A\cap B$ and $B\cap A$ are identical. The intersection of two sets can be represented by a Venn diagram as shown in Figure 3.

Figure 3 :

Example 4

Given $A=\left\{3,4,5,6\right\}$ , $B=\left\{3,5,9,10,15\right\}$ and $C=\left\{4,6,10\right\}$ state

1. $A\cap B$ ,  
2. $B\cap C$   and draw a Venn diagram representing these intersections.

Solution

1. The elements common to both $A$ and $B$ are 3 and 5. Hence $A\cap B=\left\{3,5\right\}$
2. The only element common to $B$ and $C$ is 10. Hence $B\cap C=\left\{10\right\}$

   Figure 4

   

Task!

Given $D=\left\{a,b,c\right\}$ and $F=\left\{\text{the entire alphabet}\right\}$ state $D\cap F$ .

Answer

The intersection of three or more sets is possible, and is the subject of the next Example.

Example 5

Given $A=\left\{0,1,2,3\right\}$ , $B=\left\{1,2,3,4,5\right\}$ and $C=\left\{2,3,4,7,9\right\}$ state

1. $A\cap B$
2. $\left(A\cap B\right)\cap C$
3. $B\cap C$
4. $A\cap \left(B\cap C\right)$

Solution

1. The elements common to $A$ and $B$ are 1, 2 and 3 so $A\cap B=\left\{1,2,3\right\}$ .
2. We need to consider the sets $\left(A\cap B\right)$ and $C$ . $A\cap B$ is given in (1). The elements common to $\left(A\cap B\right)$ and $C$ are 2 and 3. Hence $\left(A\cap B\right)\cap C=\left\{2,3\right\}$ .
3. The elements common to $B$ and $C$ are 2, 3 and 4 so $B\cap C=\left\{2,3,4\right\}$ .
4. We look at the sets $A$ and $\left(B\cap C\right)$ . The common elements are 2 and 3. Hence $A\cap \left(B\cap C\right)=\left\{2,3\right\}$ .

   Note from (2) and (4) that here $\left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)$ .

The example illustrates a general rule. For any sets $A$ , $B$ and $C$ it is true that

$\left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)$

The position of the brackets is thus unimportant. They are usually omitted and we write $A\cap B\cap C$ .

Suppose that sets $A$ and $B$ have no elements in common. Then their intersection contains no elements and we say that $A$ and $B$ are disjoint sets. We express this as

$A\cap B=\varnothing$

Recall that $\varnothing$ is the empty set. Disjoint sets are represented by separate area regions in the Venn diagram.

3.2 Union

The union of two sets $A$ and $B$ is a set which contains all the elements of $A$ together with all the elements of $B$ . We write $A\cup B$ to denote the union of $A$ and $B$ . We can describe the set $A\cup B$ formally by:

Thus the elements of the set $A\cup B$ are those quantities $x$ such that $x$ is a member of $A$ or a member of $B$ or a member of both $A$ and $B$ . The deeply shaded areas of Figure 5 represents $A\cup B$ .

Figure 5(a)

Figure 5(b)

In Figure 5(a) the sets intersect, whereas in Figure 5(b) the sets have no region in common. We say they are disjoint .

Example 6

Given $A=\left\{0,1\right\}$ , $B=\left\{1,2,3\right\}$ and $C=\left\{2,3,4,5\right\}$ write down

1. $A\cup B$
2. $A\cup C$
3. $B\cup C$

Solution

1. $A\cup B=\left\{0,1,2,3\right\}$
2. $A\cup C=\left\{0,1,2,3,4,5\right\}$
3. $B\cup C=\left\{1,2,3,4,5\right\}$ .

Recall that there is no need to repeat elements in a set. Clearly the order of the union is unimportant so $A\cup B=B\cup A$ .

Task!

Given $A=\left\{2,3,4,5,6\right\}$ , $B=\left\{2,4,6,8,10\right\}$ and $C=\left\{3,5,7,9,11\right\}$ state

1. $A\cup B$
2. $\left(A\cup B\right)\cap C$
3. $A\cap B$
4. $\left(A\cap B\right)\cup C$
5. $A\cup B\cup C$

Answer

Exercises

1. Given a set $A$ , its complement $A^{\prime }$ and a universal set $S$ , state which of the following expressions are true and which are false.

   (a)   $A\cup A^{\prime }=S$	(b)   $A\cap S=\varnothing$	(c)   $A\cap A^{\prime }=\varnothing$
   (d)   $A\cap A^{\prime }=S$	(e)   $A\cup \varnothing =S$	(f)   $A\cup \varnothing =A$
   (g)   $A\cup \varnothing =\varnothing$	(h)   $A\cap \varnothing =A$	(i)   $A\cap \varnothing =\varnothing$
   (j)   $A\cup S=A$	(k)   $A\cup S=\varnothing$	(l)   $A\cup S=S$

2. Given $A=\left\{a,b,c,d,e,f\right\}$ , $B=\left\{a,c,d,f,h\right\}$ and $C=\left\{e,f,x,y\right\}$ obtain the sets:

   (a)   $A\cup B$	(b)   $B\cap C$	(c)   $A\cap \left(B\cup C\right)$
   (d)   $C\cap \left(B\cup A\right)$	(e)   $A\cap B\cap C$	(f)   $B\cup \left(A\cap C\right)$

3. List the elements of the following sets:
   1. $A=\left\{x:x\text{ is odd and }x\text{ is greater than 0 and less than 12}\right\}$
   2. $B=\left\{x:x\text{ is even and }x\text{ is greater than 19 and less than 31}\right\}$
4. Given $A=\left\{5,6,7,9\right\}$ , $B=\left\{0,2,4,6,8\right\}$ and $S=\left\{0,1,2,3,4,5,6,7,8,9\right\}$ list the elements of each of the following sets:

   (a)   $A^{\prime }$	(b)   $B^{\prime }$	(c)   $A^{\prime }\cup B^{\prime }$
   (d)   $A^{\prime }\cap B^{\prime }$	(e)   $A\cup B$	(f)   ${\left(A\cup B\right)}^{\prime }$
   (g)   ${\left(A\cap B\right)}^{\prime }$	(h)   ${\left(A^{\prime }\cap B\right)}^{\prime }$	(i)   ${\left(B^{\prime }\cup A\right)}^{\prime }$

   What do you notice about your answers to (c),(g)?

   What do you notice about your answers to (d),(f)?

5. Given that $A$ and $B$ are intersecting sets, i.e. are not disjoint, show on a Venn diagran the following sets
   1. $A^{\prime }$
   2. $B^{\prime }$
   3. $A\cup B^{\prime }$
   4.   $A^{\prime }\cup B^{\prime }$
   5. $A^{\prime }\cap B^{\prime }$

Answer