Set Operations

A set includes objects, and each object is called an element. The operations that are applied to two or more sets to develop a relationship between them are called set operations. There are four types of set operations:

• Union of sets

• Intersection of sets

• Set difference

• Complement of sets

A Venn diagram is a visual aid used to express relationships between sets. Given below is a Venn diagram for reference:

Set operations are rules that we use to compare and combine different sets of data. The four basic operations are:

• Union of sets

• Intersection of sets

• Set difference

• Complement of sets

Combining all elements from both sets without repeating any element is known as the union. Let’s consider two sets, A and B.

Their union is represented as A ⋃ B. We can calculate the number of elements in A ⋃ B by using the formula n(A ⋃ B) = n(A) + n(B) — n(A ∩ B). Here, n(X) indicates how many elements the set X has. 

Example: 

Let A = {apple, banana}

B = {banana, mango}

Then A ⋃ B = {apple, banana, mango}

Intersection refers to the things that are common to both sets. The intersection of set A and B is denoted by A ∩ B. The number of elements in the intersection is derived from the union formula and given by n(A ∩ B) = n(A) + n(B) - n(A ⋃ B). To understand the intersection of sets, refer to the Venn diagram given below:

Example: 

A = {cat, dog, rabbit}

B = {dog, rabbit, parrot}

A ∩ B = {dog, rabbit}

The set difference represents the elements that are in set A but not in set B. It is written as A – B, which means all elements in A that are absent in B. To understand the set difference better, let’s see an example.

Let A = {red, blue, green}

B = {blue, yellow}

A – B = {red, green}

The complement of a set means all the elements belong to the universal set but not in the given set. The complement of a set can be represented as A’ or Ac. 

Example: 

Let the universal set U = {1, 2, 3, 4, 5, 6}

A = {1, 2, 3}

Then, A’ = {4, 5, 6}

The properties of set operations are certain rules that we must follow while performing operations like union, intersection, and complement on two or more sets. The following properties help us to understand and simplify problems in set theory. 

1. Commutative Law

Commutative law states that when applying union or intersection to sets, the order of the sets doesn’t have any effect on the outcome.

For any two sets A and B, the commutative property is given as: A ⋃ B = B ⋃ A

This means that combining two sets gives the same result, no matter which set comes first.
A ∩ B = B ∩ A

This means the common elements in the sets are the same regardless of the order. 

2. Associative Law

Associative property shows that the grouping of sets doesn’t change the result in the union and intersection.
(A ⋃ B) ⋃ C = A ⋃ (B ⋃ C)

We can group the sets in any order when taking their union. 
(A ∩ B) ∩ C = A ∩ (B ∩ C)
When finding common elements, we can group the sets in any way.

3. De Morgan’s Law

De Morgan’s law tells us how complements work with union and intersection. 
(A ⋃ B)’ = A’ ∩ B’

The elements not in A or B are the same as those not in A and not in B.
(A ∩ B)’ = A’ ⋃ B’

The complement of the intersection of A and B has all elements that are not shared by both sets and belong to either A or B, but not both. 

4. Identity and Null Set Properties

Joining a set with itself gives the same set,
A ⋃ A = A

The common elements in a set with itself are the same set,
A ∩ A = A

There are no common terms between any set and the empty set.
A ∩ ∅ = ∅

Adding an empty set to a set doesn’t change the set
A ⋃ ∅ = A

5. Subset Properties

The common part of A and B will always be a part of A.
A ∩ B ⊆ A

A will always be a part of the union of A and B.
A ⊆ A ⋃ B

Sets are represented using curly braces {} to group their elements. The elements in the set can be anything, like numbers, letters, or other sets. The common ways to represent sets are:

• Roster Form

• Set-Builder Form

• Interval Notation

• Special Sets

If the elements are written within the curly braces and separated by commas is known as the roster form. A = {1, 2, 3, 4, 5} is a common example of a roster form.

The way of describing a set without listing all the elements is known as a set builder form. Instead of listing all the elements in a set, we describe a property that describes the elements present in the set. The notation of a set-builder form is {x | P(x)}, where x represents the elements and P(x) is the condition.

For example, A = {x | x is an odd number}; this means that the elements in the set A are odd, like 1, 3, 5, 7, etc.

Intervals are used to represent a group of numbers on the number line, especially real numbers. In set notation, we can use intervals to represent a range of values instead of listing each element individually. Let’s consider an example to understand the interval notation: B = {x | 0 < x < 5} can be represented as (0, 5), indicating all the real numbers greater than 0 and less than 5.

Empty set: The set with no elements is known as the empty set. It is denoted as ∅ or {}.

Universal set: The set that contains all elements under consideration is known as a universal set and is denoted by U.

Singleton set: A set containing only one element is called a singleton set.

In sets, elements are listed inside the curly brackets {} and separated by commas. Each element in the set will be unique, and no duplicate elements are allowed inside a set. Elements can be in any form, such as numbers, letters, symbols, or other sets.

Example: A = {7, 8, 9, 5}

Here, 7, 8, 9, and 5 are the elements of a set A.

Each element is unique and not repeated inside a set.

The order of the elements does not matter, as they contain the same elements.

A set can look different depending on what it contains and how they are grouped. Some sets have no element at all, and some sets have more elements; some may be a part of other sets, and some may have the same elements. Learning the types of sets helps us compare, organize, and solve problems easily in set theory.

Null or Empty or Void Set: A set with no elements is called a null or empty, or void set. The null set can be denoted by ∅ or {}.

Singleton Set: A set consisting of a single element is called a singleton set.

Finite Set: The set that has only a finite number of elements is known as a finite set.

Infinite Set: A set with an infinite number of elements.

Subset: All the elements from set A are found in set B; then A is termed a subset of B. The subset can be denoted as A ⊆ B.

Proper Subset: When A is a subset of B, but A is not equal to B, then A is considered a proper subset of B. A proper subset can be represented as A ⊂ B, where A ≠ B.

Universal Set: The set that consists of all the elements that occur in the discussion is known as a universal set. The universal set can be denoted as U.

Power Set: If A is the given set and all the subsets of A are called the power set of A and are denoted as P(A).

Equal Set: If every element of the set A is also an element of set B, or vice versa, then it is called an equal set. Equal sets are represented as A = B.

Disjoint Sets: If two sets do not have any common elements, it is known as disjoint sets. Disjoint sets are denoted as A ∩ B = ∅. This denotes that the intersection of disjoint sets results in a null set or a set with no elements. 

Set formulas are mathematical expressions that show how different operators relate to each other. Learning the set formulas makes it easier to solve problems in algebra, computer science, probability, and more. Listed below are some of the important set formulas:

• n(A ⋃ B) = n(A) + n(B) - n(A ∩ B)

• n(A ⋃ B) = n(A) + n(B) if and only if A, B are disjoint sets.

• n(A – B) + n(A ∩ B) = n(A)

• n(B – A) + n(A ∩ B) = n(B)

• n(A – B) + n(A ∩ B) + n(B – A) = n(A ⋃ B)

• n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

We use sets and set operations in many places. They are used not only in math but also in daily life, such as computers, science, and more. Here are some of the real-life applications of set operations.

• Education: In schools, set operations help teachers to group students based on their interests or performance. Set operations can also be used for planning lessons, assigning group projects, or activities. It is also used to compare class performance and analyze test results.

• Computer Science: Set operations are used in search engines, apps, and databases. They help computers match data, sort users, filter information, and manage files and emails, which makes computers faster and smarter by handling a large amount of data.

• Science: Scientists use set operations to study and group different kinds of plants, animals, or chemical elements. It also helps in finding common features, comparing different groups, and separating unique characteristics for research.