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# Operations on Sets

We know that we can perform basic operations on whole numbers, decimals, fractions, etc. But did you know that we can also perform operations on sets?

The operations we perform on sets are not the same addition, subtraction, multiplication, and division operations that we perform on the above types of numbers. Instead, there are four types of operations we perform on sets. These are:

1. Intersection
2. Union
3. Difference
4. Complement

Let's look into each of these operations in detail and learn the symbols used, the purpose of each operation, and the results each operation has on sets.

## Union of sets

Suppose we have two sets, A and B. The union of sets A and B is a new set that contains all the elements present in both sets A and B. This subset is denoted by the following symbol:

$A\cup B=\left\{x:x\in Aorx\in B\right\}$

where x represents the elements in both sets A and B. ∈ is the symbol that means "element of".

Example 1

Set A: $\left\{1,2,3\right\}$

Set B: $\left\{4,5,6,7\right\}$

Set A ∪ Set B: $\left\{1,2,3,4,5,6,7\right\}$

Example 2

Set A = $\left\{A,C,E,G\right\}$

Set B = $\left\{B,D,F,H\right\}$

Set A ∪ Set B =

Example 3

Sets do not necessarily have to "fall in order" as these two sets have. We can have the following:

Set A: $\left\{1,3,8,10\right\}$

Set B: $\left\{12,13,17,19\right\}$

Set A ∪ Set B: $\left\{1,3,8,10,12,13,17,19\right\}$

## Intersection of sets

Say we have two sets, A and B. The intersection of sets A and B is the subset of the universal set U, which is made up of the elements that are common to both sets A and B. The intersection of sets is indicated by the symbol '∩'. The operation is denoted as:

$A\cap B=\left\{x:x\in Aandx\in B\right\}$

where x equals the common element of both sets A and B.

The intersection of sets A and B can also be denoted as:

$A\cap B=n\left(A\right)+n\left(B\right)-n\left(A\cup B\right)$

where:

$n\left(A\right)$ is the cardinal number of set A,

$n\left(B\right)$ is the cardinal number of set B, and

$n\left(A\cap B\right)$ is the cardinal number of the intersection of set A and B.

Example 4

Set A: $\left\{A,B,C,D,E\right\}$

Set B: $\left\{D,E,F,G,H\right\}$

Set $A\cap B:\left\{D,E\right\}$ because only D and E are present in both sets A and B.

Example 5

Set A: $\left\{1,1,2,3,5,8,13,21,34,55,89\right\}$

Set B: $\left\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97\right\}$

Set $A\cap B:\left\{2,3,5,13,89\right\}$

Note: In this case, Set A represents the Fibonacci numbers from 1-100, and Set B represents the prime numbers from 1-100. The intersection of Set A and Set B, denoted as Set $A\cap B$ , contains all the numbers that are both Fibonacci numbers and prime numbers from 1-100. This example illustrates how the intersection of sets can be used to find relevant information in real-world scenarios.

Example 6

It's possible to run an intersection of sets operation on more than 2 sets. For example, say a couple has three daughters and they are curious to know how many letters in their names are similar across all three children.

Set A: $\left\{J,E,N,N,I,F,E,R\right\}$

Set B: $\left\{J,O,C,E,L,Y,N\right\}$

Set C: $\left\{J,A,M,I,E,L,Y,N,N\right\}$

Set $A\cap B\cap C=\left\{J,E,N\right\}$

## Difference of sets

Again, we have two sets, A and B. The difference of sets A and B is equal to the set that consists of elements present in set A without those in set B. It is denoted as:

$A-B$

In other words, we could say that the difference of set A and B is equal to the intersection of set A with the complement of set B. Therefore,

$A-B=A\cap B$ .

Example 7

Find the set that is the difference of sets $A-B$

Set A: $\left\{10,11,12,13,14,15,16,17\right\}$

Set B: $\left\{14,15,16,17\right\}$

So the difference of the sets, or set $A-B$ , is:

Set A - B: $\left\{10,11,12,13\right\}$

## Complement of set

If U is a universal set and x is any subset of U, then the complement of x is the set of all the elements of the set U apart from the elements of x.

We denote this as:

$x=\left\{a:a\in Uanda\notin A\right\}$

The complement set of set A is called set ${A}^{C}$ .

Example 8

$U=\left\{1,2,3,4,5,6,7,8,9,10\right\}$

$A=\left\{1,3,5,7,9\right\}$

${A}^{C}=\left\{2,4,6,8,10\right\}$

## Properties of set operations

Commutative property

$A\cup B=B\cup A$

$A\cap B=B\cap A$

Associative property

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

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

Distributive property

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

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

Subsets

## Flashcards covering the Operations on Sets

Statistics Flashcards

Set Theory Flashcards

## Get help learning about operations on sets

Tutoring is an excellent way for your student to learn all the ins and outs of performing operations on sets. After they get used to operations on numbers, it can be confusing to have to switch to doing operations on sets, which are quite different. There may not be enough time offered in the classroom in order for your student to gain a thorough understanding of the operations on sets. That's where a private tutor comes in. A tutor can sit with your student as they complete problems involving operations on sets and guide them, making sure they do the operations correctly the first time and don't develop bad habits. The 1-on-1 attention offered by a tutor could be exactly what your student needs to catch on to operations on sets.

If you'd like us to set your student up with a tutor that matches their needs, contact Varsity Tutors today and speak with one of our helpful Educational Directors. We look forward to hearing from you and helping your student.

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