Last updated on July 5th, 2025
Properties of sets enable the simplification of set operations, which include union, intersection, and complement operations. The set has properties like the commutative and associative properties, which work similarly to how they do with real numbers. Let us learn about the properties of sets.
A set is a collection of well-defined objects. A set is represented by using capital letters. Each object present in a set is called an element. The elements in the set are enclosed by curly braces - { }. A set can contain various types of items, such as objects, people, numbers, or shapes. For example, a set of even numbers can be represented as E = { 2, 4, 6, 8, …}
Sets follow certain properties that make operations like union, intersection, and complement easier to work with. Some of these properties, such as the commutative and associative properties, are similar to the ones we use in arithmetic and algebra. The following are some of the key properties of sets:
Property |
Formula |
Description |
Commutative property |
A ∪ B = B ∪ A |
The order of union or intersection does not change the result. |
Associative Property |
(A ∪ B) ∪ C = A ∪ (B ∪ C) |
The grouping of sets does not affect the outcome. |
Distributive Property |
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
Union distributes over the intersection, and intersection distributes over the union. |
Identity Property |
A ∪ ∅ = A |
The empty set acts as an identity for union, and the universal set acts as an identity for intersection. |
Idempotent Property |
A ∪ A = A |
The union or intersection of a set with itself results in the same set. |
Complement Laws |
A ∪ A′ = U |
A set and its complement together form the universal set, and their intersection is the empty set. |
The union of sets is created by bringing together all the elements from two or more sets. It is represented by the symbol “∪”. When combining the sets, any elements that appear in more than one set are written only once. Here are some important properties of the union of sets:
Properties Of the Intersection Of Sets
The intersection of sets refers to the elements that are shared between two sets. It is shown by the symbol “∩”. If the sets have no elements in common, their intersection is an empty set, which is represented by “∅”. The key properties of the intersection of sets include:
Commutative Law: A ∩ B = B ∩ A
Associative Law: (A ∩ B) ∩ C = A ∩ (B ∩ C)
Identity Property: U ∩ A = A
Idempotent Property: A ∩ A = A
Dominant Property: A ∩ ∅ = ∅
Distributive Property: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Properties Of Complement Sets
The complement of a set includes all the elements that are not part of that set. If we have a set A, then the complement of a set A is written as A′. The key properties of complement sets include:
Complement Laws: A ∪ A′ = U (over union)
A ∩ A′ = ∅ (over intersection)
Double Complement Law: (A′)′ = A
Universal Set Complement: U′ = ∅
Empty Set Complement: ∅′ = U
De Morgan’s Laws: (A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′
As we know, a set is a collection of well-defined objects. The properties of sets are used in many scenarios in real life, such as for arranging items like books, music, and clothes.
When students are learning concepts like union, intersection, and complement of sets, they can sometimes get confused by the symbols, meanings, or how to use the properties correctly. Let us see some mistakes and how they can be avoided.
Prove that A ∪ B = B ∪ A by using the sets A = {2, 3, 4} and B = {6, 7, 8}.
A ∪ B = {2, 3, 4} ∪ {6, 7, 8} = {2, 3, 4, 6, 7, 8}
B ∪ A = {6, 7, 8} ∪ {2, 3, 4} = {2, 3, 4, 6, 7, 8}
The union of two sets combines the elements in the two sets. As the A ∪ B and B ∪ A result in the same set, this proves the commutative property, i.e., A ∪ B = B ∪ A.
Find the intersection of the given sets A = {3, 6, 9} and B = {3, 6, 8, 9}.
A ∩ B = {3, 6, 9} ∩ {3, 6, 8, 9} = {3, 6, 9} = A
The intersection of two sets shows the elements that both sets have in common. The sets A and B share the numbers 3, 6, and 9, then their intersection is A ∩ B = {3, 6, 9}.
Show that A ∪ A = A and A ∩ A = A for the set A = {x, y, z}.
A ∪ A = {x, y, z} ∪ {x, y, z} = {x, y, z}
A ∩ A = {x, y, z} ∩ {x, y, z} = {x, y, z}
The idempotent law means that if you take the union or intersection of a set with itself, you will get the same setback. This is because we are not adding any new elements. So, A ∪ A = A and A ∩ A = A.
Find the union of the given sets A = {4, 5, 6, 7, 8} and B = {3, 5, 9,10}.
A ∪ B = {4, 5, 6, 7, 8} ∪ {3, 5, 9,10} = {3, 4, 5, 6, 7, 8, 9, 10}
The union of two sets means combining all the elements from both sets. The set A has the elements 4, 5, 6, 7, and 8, and set B has 3, 5, 9, and 10, combining them gives us A ∪ B = {3, 4, 5, 6, 7, 8, 9, 10}.
Given the set S = {8, 9, 12}. Find A ∪ ∅.
A ∪ ∅ = {8, 9, 12} ∪ ∅ = {8, 9, 12}.
The identity property of union states that the union of any set with the empty set results in the set itself. Hence, A ∪ ∅ = {8, 9, 12}.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.