Set Operations

A set is simply a collection of distinct objects. These objects can be numbers, letters, or even people—anything! We denote a set using curly brackets.
For example: A = {1, 2, 3}

Set Operations can be defined as the operations performed on two or more sets to obtain a single set containing a combination of elements from all the sets being operated upon.

• Set operations are mathematical operations operated on sets, which are collections of distinct objects or elements.
• There are three major types of operation on sets: Union (∪), Intersection (∩), and Difference (-).
  Other operations include Difference, Complement, Addition, & Subtraction.

The most common set operations, such as union, intersection, disjoint, set difference, etc., will be explored in detail below, including their definitions, examples, and Venn diagrams.

Union of Sets

The Union of sets A and B, denoted by A ∪ B, is the set of distinct elements that belong to set A or set B, or both. This operation can be represented as;

A ∪ B = {x: x ∈ A or x ∈ B}

In simple terms, the union includes every element that appears in either of the two sets, without any repetition. Where x is the elements present in both sets A and B.

Venn Diagram For Union of Sets

The area shaded in green represents A ∪ B or the union of sets A and B.

Example: Find the union of A = {2, 3, 4} and B = {3, 4, 5}.
Solution:

A ∪ B = {2, 3, 4, 5}.

Intersection

The intersection of the sets A and B, denoted by A ∩ B, is the set of elements that belong to both A and B, i.e. set of the common elements in A and B. This operation is represented as:

A∩B = {x: x ∈ A and x ∈ B}

In other words, the intersection contains only those elements that are present in both sets. Here, x represents the elements that are common to both sets A and B.

Venn Diagram For Intersection of Sets

The area shaded in green represents A∩B or the intersection of sets A and B, which includes the elements common to both sets A and B.

Example: Find the intersection of A = {2, 3, 4} and B = {3, 4, 5}

Solution:

Look for elements that are common to both sets A and B.

A ∩ B = {3, 4}.

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

A ∩B = n (A) + n (B) – n (A∪B)

Where,

• n(A)= the cardinality (or number of elements) of set A,
• n(B)= the cardinality of set B,
• n(A∪B) = the cardinality of the union of sets A and B.

Disjoint Set

Two sets are said to be disjoint if their intersection is the empty set. i.e., sets have no common elements. In simpler terms, they don’t “overlap” at all.
So if you try to find their intersection, you’ll get the empty set, which we denote by the symbol ϕ or {}.

Venn Diagram For Disjoint Sets

The sets A and B are disjoint, meaning they have no common elements (no overlap).

For Example: Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}

Solution:

A and B are disjoint sets since both of them have no common elements.

Set Difference 

The difference between sets is denoted by 'A - B', which is the set containing elements that are in A but not in B i.e., all elements of A except the element of B.

Venn Diagram For Set Difference

In the below diagram, the set difference A−B contains all the elements that are in A but not in B.

Example: If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}, find A - B.
Solution:

A - B = {1, 3, 5}

We can also state that the difference between set A and set B is equivalent to the intersection of set A with the complement of set B. Therefore,

A − B = A ∩ B′

where B′ = the complement of set B.

Read More: Set Difference

Symmetric Difference

The symmetric difference of A and B includes elements in A or B but not both.

• It is denoted by: A △ B or A ⊕ B.
• The symmetric difference is like saying, “Give me everything that’s not shared.
• It is defined as: A △ B = (A−B) ∪ (B−A)

Venn Diagram For Symmetric Difference

The symmetric difference AΔB includes elements that are in either A or B but not in both.

Example: Let set A = {1, 2, 3}, and set B = {3, 4, 5}, then Find the symmetric difference

Solution:

A △ B = {1, 2, 4, 5}

Complement of a Set

If U is a universal set and X is any subset of U, then the complement of X consists of all the elements in U that are not in X.

X^' = {a : a∈ U and x ∉ A}

Venn Diagram For Complement of a Set

In the diagram below, set A^' includes all elements not in A, relative to the universal set.

Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8} And A = {1, 2, 5, 6}

Solution:

Then, the complement of A, denoted as A′, will be: A′ = {3, 4, 7, 8}

Read More: Complement of a Set

Addition and Subtraction 

The addition of sets A and B, known as Minkowski addition, results in a new set where each element is the sum of every possible pair of elements, one from set A and one from set B. In other words, the new set contains all possible sums of elements taken from the two sets. Set subtraction operates similarly but involves subtracting each element of set B from each element of set A. These operations are only meaningful when applied to numeric data types; if applied otherwise, they become symbolic with no practical significance.

Additionally, set addition is commutative, meaning the order of the sets does not affect the result. However, set subtraction is not commutative, as the order of the sets does affect the outcome.

n (A ∪ B) = n(A) + n(B) - n (A ∩ B)

A - B = A ∩ B'

Properties of Set Operations

Set operations have several important properties that govern their behavior. Here are some fundamental properties of set operations:

Closure Property

• Set operations are closed under their respective operations, meaning that operating on sets results in another set.
• For example, the union, intersection, and difference of sets always produce sets as their results.

Commutative Property

• Union: A ∪ B = B ∪ A
• Intersection: A ∩ B = B ∩ A
• Symmetric Difference: A Δ B = B Δ A

Associative Property

• Union: (A ∪ B) ∪ C = A ∪ (B ∪ C)
• Intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Property

• Union over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• Intersection over Union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Identity Property

• Union: A ∪ ∅ = A
• Intersection: A ∩ U = A, where U represents the universal set
• Symmetric Difference: A Δ ∅ = A

Complement Property

• Union: A ∪ A' = U, where U is the universal set
• Intersection: A ∩ A' = ∅ (the empty set)

Absorption Property

• Union over Intersection: A ∪ (A ∩ B) = A
• Intersection over Union: A ∩ (A ∪ B) = A

Solved Question of Operations on Sets

Question 1 : Find the union of two sets A = {8, 10, 14} and B = {7, 16}

Solution:

(A ∪ B) = {8, 10, 14} ∪ {7, 16}
(A ∪ B) = {7, 8, 10, 14, 16}

Question 2: Find the intersection of sets P = {a, n, x} and Q = {x, y, z}

Solution:

(P ∩ Q) = {a, n, x} ∩ {x, y, z}
(P ∩ Q) = {x}

Question 3: Find the complement of set X = {4, 6, 9} where Universal set U = {1, 2, 3, 4, 6, 9}

Solution:

X' = U - X
X' = {1, 2, 3, 4, 6, 9} - {4, 6, 9}
X' = {1, 2, 3}

Question 4: Given two sets A = {5, 6, 9, 10} and B = {3, 6, 12} then find, A - B and B - A

Solution:

A - B = {5, 6, 9, 10} - {3, 6, 12}
A - B = {5, 9, 10}
B - A = {3, 6, 12} - {5, 6, 9, 10}
B - A = {3, 12}

Question 5: Find the number of elements in the set (A ∪ B) given that n(A) = 10, n(B) = 4,, and n (A ∩ B) = 5.

Solution:

To find n (A ∪ B) we use formula:
n (A ∪ B) = n(A) + n(B) - n (A ∩ B)
n (A ∪ B) = 10 + 4 - 5
n (A ∪ B) = 9

Practice Questions on Set Operations

Question 1. Find the union of two sets A = {2, 4, 5, 9} and B = {2, 6, 10, 12}.

Question 2. Find the intersection of sets P = {5, 6, 7, 9} and Q = {6, 9}.

Question 3. Find the complement of set X = {p, q, r} where Universal set U = {e, f, h, p, q, r}.

Question 4. Given two sets A = {1, 2, 4, 7, 10} and B = {1, 2, 8, 15}, then fin, B - A.

Question 5. Find the number of elements in set A, given that n (A ∪ B) = 12, n(B) = 7, and n (A ∩ B) = 4.

Related Articles:

• Set Theory
• Set Symbols
• Types of Sets