102 LearnersLast updated on July 5th, 2025
Properties of Sets
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.
What is a set?
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, …}
What are the properties of sets?
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:
- Commutative property
- Associative property
- Distributive property
- Complement property
- Identity property
- Idempotent property
Property
Formula
Description
Commutative property
(Union and Intersection)A ∪ B = B ∪ A
A ∩ B = B ∩ AThe order of union or intersection does not change the result.
Associative Property
(Union and Intersection)(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)The grouping of sets does not affect the outcome.
Distributive Property A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)Union distributes over the intersection, and intersection distributes over the union.
Identity Property
A ∪ ∅ = A
A ∩ U= AThe empty set acts as an identity for union, and the universal set acts as an identity for intersection.
Idempotent Property
A ∪ A = A
A ∩ A = AThe union or intersection of a set with itself results in the same set.
Complement Laws
A ∪ A′ = U
A ∩ A′ = ∅A set and its complement together form the universal set, and their intersection is the empty set.
Properties Of the Union Of Sets
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:
- Commutative Law: A ∪ B = B ∪ A
- Associative Law: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Identity Property: A ∪ ∅ = A
- Idempotent Property: A ∪ A = A
- Dominant Property: U ∪ A = U
- Distributive Property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
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′
Real-Life Applications on Properties of Sets
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.
- Library and Information Management: For sorting and organizing books efficiently, librarians use principles of set theory. For example, to group books under the genre of horror, romance, thriller, fiction, library management uses a union of categories for easy identification. For readers also, it is easier to access books.
- Music and Digital Playlists: For organizing songs or podcasts, music streaming platforms use set theories to suggest a playlist based on preferences. These streaming platforms also curate an unique playlist without repetition by using a union of sets. Even also add those songs in the playlist that come under the intersection of the playlists.
- Social Media Platforms: Many websites and apps use set theory to connect users and organize content. For example, using the intersection of sets, they can find people who have similar interests or mutual friends. By using these ideas, they can give each person a more personalized and enjoyable experience.
Common Mistakes and How to Avoid Them in Properties of Sets
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.
Mistake 1
Confusing Union with Intersection
Students often mix up the meanings of union and intersection. They might think that union means just the shared elements, and intersection means putting all elements together. For example, they may have a misconception that A ∪ B result in common elements instead of thinking that the union (∪) means combining. Students should practice more about symbols and their definitions.
Mistake 2
Repeating Elements in a Set
Sometimes students list the same element more than once in a set. For example, they might write the set of factors of 4 as {1, 2, 2, 4}, which is incorrect. Sets are supposed to contain unique elements. They should be taught that each element in a set should appear only once.
Mistake 3
Incorrect Use of Curly Braces
tudents can get confused about which brackets to use when writing sets. Instead of curly braces { }, they might use parentheses () or square brackets [ ]. They should be taught through using examples that curly braces {} are the correct notation for sets.
Mistake 4
Thinking That Sets Are Always Finite
Some students think that sets always have a fixed number of elements. For example, when writing the set of natural numbers, instead of recognizing that the set continues into infinitely. Students should practice more examples of infinite sets, like the set of natural numbers and the set of whole numbers.
Mistake 5
Miscounting Elements in a Set
Students may count the number of elements in a set incorrectly. For example, they might skip the elements while counting the elements in the set. Practicing how to carefully count the distinct elements will help them get more accurate at determining the size of a set.
Solved examples of Properties of Sets
Problem 1
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}
Explanation
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.
Problem 2
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
Explanation
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}.
Problem 3
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}
Explanation
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.
Problem 4
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}
Explanation
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}.
Problem 5
Given the set S = {8, 9, 12}. Find A ∪ ∅.
A ∪ ∅ = {8, 9, 12} ∪ ∅ = {8, 9, 12}.
Explanation
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}.
FAQs on Properties of Sets
1.What are the fundamental properties of set operations?
2.What is the identity property in set operations?
3.What is the union of {2, 3, 4} and ∅?
4.What is the absorption law in set theory?
5.What are the applications of set properties?
6.How can children in United States use numbers in everyday life to understand Properties of Sets?
7.What are some fun ways kids in United States can practice Properties of Sets with numbers?
8.What role do numbers and Properties of Sets play in helping children in United States develop problem-solving skills?
9.How can families in United States create number-rich environments to improve Properties of Sets skills?
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Hiralee Lalitkumar Makwana
About the Author
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.
Fun Fact
: She loves to read number jokes and games.
