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# Sets and Set Difference

## Introduction to Sets

Last updated date: 19th Mar 2023
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In addition to mathematics, sets are used frequently in everyday life. The most practical example of a set is a kitchen. The kitchen is usually well organized by our mother. Kids’ school bags are another example, where you keep all your copies and books. Together make a bag, and we can call it a set. One of the most significant and fundamental sets theory operations is set difference. The difference between two sets, A and B, is another set consisting of elements from A that are NOT in B. In this article, we will learn about the difference of 2 sets and some of the properties of set difference.

## Sets

In mathematics, a set is a well-defined collection of objects. Sets are named and represented using capital letters. In set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.

From the below image, we can say that there is the P set in which different shapes are there. And it can be termed as subsets.

Pictorial Understanding of Sets

## Set Difference

The set operation difference between sets implies subtracting the elements from a set, similar to the difference between numbers. The difference between sets $A$ and $B$, denoted as $A-B$, lists all the elements in set $A$ but not in set $B$. $A-B$ states that elements of $A$ are not the elements of $B$. Therefore, $A - B = \{x : x \in A$, and $x \notin B\}$ and $B-A=\{x: x \in B$, and $x \notin A\}$

## Example of AB Sets

So, as far as we have learned, the sets and sets are different. Now let’s see what AB sets are. If A and B are two sets and each element of set A is also an element of set B, then A is referred to as a subset of B.

For example,

Let A = {1, 3, 5}

B = {5, 1, 7, 3}

Here, A is a subset of B.

Since all the elements of set A are contained in set B,

But B is not a subset of A, as all the elements of set B are not contained in set A.

Set Difference

## Set Difference Properties

Set difference properties are classified into two categories:

1. The set remains unchanged by the change in the order in which the components are written. Hence, a set {x,y,z} can be write as {y,z,x} or {y,x,z} or {x,z,y} or {z,y,x} or {z,x,y}

For example A={3,6,9}

Similarly, {3,6,9} = {6,9,3} = {3,9,6} = {9,6,3} = {9,3,6}

1. The set remains the same if any element is repeated more than once. To be more precise, repeated elements are considered as a single element, i.e. {1,2,2,2,3,4,4} ={1,2,3,4}

## Difference Between 2 Sets

• If A and B are two sets, then to find their difference, it is needed to write in A - B or B - A.

• If A = {2, 3, 4} and B = {4, 5, 6}

A - B describes the elements of A that are not B's elements.

i.e., in the above example, A - B = {2, 3}

In general, A - B = {x : x ∈ A, and x ∉ B}

• If A and B are disordered sets, A - B equals A, and B - A equals B.

An Example of a Set Difference:

Find the differences between 2 sets A-B and B-A, where set A = {1, 2, 3, 4} and set B = {2, 3, 5, 7}

Ans: It is given that A = {1, 2, 3, 4} and B = {2, 3, 5, 7}. Then

A - B = 1, 2, 3, 4- 2, 3, 5, 7 = 1, 4,

B - A = 2, 3, 5, 7-1, 2, 3, 4 = 5, 7

Therefore, A - B = {1, 4} and B - A = {5, 7}

## Solved Problems

Q 1. If $A=\{1,2,3,4,5,6\}$ and $B=\{3,4,5,6,7,8\}$, then find $A-B$ and $B-A$.

Ans: From the given question, we have the sets given as

$A=\{1,2,3,4,5,6\}$

$B=\{3,4,5,6,7,8\}$

Now finding the A - B and B - A.

$A-B=\{1,2\}$ since the elements 1,2 are there in $A$ but not in $B$.

Similarly,

$B-A=\{7,8\}$, since the elements 7 and 8 belong to $B$ and not to $A$.

Q 2. If $X=\{11,12,13,14,15\}, Y=\{10,12,14,16,18\}$ and $Z=\{7,9,11,14,18,20\}$, then find the following:

(i) $X-Y-Z$

Ans: From the given question, we have the sets given as

$X=\{11,12,13,14,15\}$

$Y=\{10,12,14,16,18\}$

$Z=\{7,9,11,14,18,20\}$

$X-Y-Z=\{11,12,13,14,15\}-\{10,12,14,16,18\}-\{7,9,11,14,18,20\}$ $=\{13,15\}$

(ii) $Y-X-Z$

Ans:$Y-X-Z=\{10,12,14,16,18\}-\{11,12,13,14,15\}-\{7,9,11,14,18,20\}$ $=\{10,16\}$

## Practice Questions

Q 1. Find the differences between sets $A-B$ and $B-A$, where set

$A=\{1,2,3,4\} \text { and set } B=\{2,3,5,7\}$

Ans: $A-B=\{1,4\}$ and $B-A=\{5,7\}$

Q 2. Let set A = {1, 2, 4, 5, 6, 7, 9}, and set B = {3, 4, 6, 7, 9} Find A - B.

Ans: A - B={1, 2, 5}

## Summary

A set is a mathematical representation of a collection of various items. It contains elements that may be any type of mathematical object. A set is always defined as a term surrounded by curly brackets. The term “set difference” subtracts one set from another. A subtraction symbol between two sets defines the difference between the sets, i.e., (A-B). The set remains the same if the order of data inside the brackets is changed, and the set value remains the same if any number comes more than once.

## FAQs on Sets and Set Difference

1. What are Universal Sets?

A universal set, denoted by the letter 'U', is the collection of all the elements regarding a particular subject.

Example: Let U = {The list of all road transport vehicles}. Here, a set of cycles is a subset of this universal set.

2. What is the meaning of the A ⊄ B symbol used in sets?

The meaning of A ⊄ B is that the left set is not a subset of the right set.

3. What are the Subsets of all the sets?

Power Set is the Subset of all the sets.