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Properties Of Sets

Properties of sets help in easily performing numerous operations across sets. The operation of union of sets, intersection of sets, complement of a set can be easily performed with the help of their respective properties. Many of the properties such as commutative property, associative property are similar to the properties of real numbers.

Let us learn more about the property of union of sets, property of intersection of sets, property of complement of sets, with the help of examples, FAQs.

1.	What Are Properties Of Sets?
2.	Properties Of Union Of Sets
3.	Properties Of Intersection Of Sets
4.	Properties Of Complement Sets
5.	Examples On Properties Of Sets
6.	Practice Questions
7.	FAQs On Properties Of Sets

What Are Properties Of Sets?

Properties of sets are the same as the properties of real numbers. Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets. are commutative property, associative property, distributive property, identity property, complement property, and idempotent property. The formulas of the properties for the, three sets A, B, and C are as follows.

Properties of Sets

The various operations of the union of sets, the intersection of sets, Complement of sets, for the given sets can be easily performed using the above properties of sets.

Properties Of Union Of Sets

The union of two sets is the new set obtained by combining and writing all the elements of the two given sets together. For the union of two sets the common elements of the two sets are not repeated and are written only once. The properties of union of sets follow the commutative law, associative law, similar to the real numbers. The important properties of union of sets is as follows.

A ∪ B = B ∪ A (Commutative law)
( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative Law)
A ∪ φ = A (Law of identity element, φ is the identity of ∪)
A ∪ A = A (Idempotent law)
U ∪ A = U (Law of U)

Properties Of Intersection Of Sets

The intersection of two sets is common element of the two sets. The number of elements in the intersection of two sets is lesser than the number of elements in the individual set. The intersection of sets follow the commutative law, associative law, distributive law, idempotent law. The important properties of the intersection of sets is as follows.

A ∩ B = B ∩ A (Commutative law).
( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
φ ∩ A = φ, U ∩ A = A (Law of φ and U).
A ∩ A = A (Idempotent law)
A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over ∪

Properties Of Complement Sets

The complement of a set is the remaining elements in the universal set, which does not belong to this set. The complement of a set A is A', and it follows the commutative law as the union and intersection of sets. The important properties of complement set are as follows.

Complement Laws: A ∪ A′ = U (ii) A ∩ A′ = φ
De Morgan's Laws: (i). (A ∪ B)´ = A′ ∩ B′
De Morgan's Laws: (ii) (A ∩ B)′ = A′ ∪ B′
Law of Double Complementation:(A')' = A
Law of Empty Set and Universal Set: φ′ = U and U′ = φ

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Examples on Properties Of Sets

Example 1: Find the union of sets A and B and prove that it follows the commutative property of the union of sets. Given A = {1, 2, 3, 4, 5, 6}, and B = {2, 3, 5, 7, 8, 9}.

Solution:

The given sets are A = {1, 2, 3, 4, 5, 6}, and B = {2, 3, 5, 7, 8, 9}.

The commutative property of union of sets is A U B = B U A.

A U B = {1, 2, 3, 4, 5, 6} U {2, 3, 5, 7, 8, 9} = {1, 2, 3, 4, 5, 6, 7, 8, 9}

B U A = {2, 3, 5, 7, 8, 9} U {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6, 7, 8, 9}

From the above two sets, we can observe that A U B = B U A

Therefore the two given sets follow the commutative property of the union of sets.
Example 2: Find the complement of a set A and prove that it follows the property of sets of the law of double complement. Given A = {1, 2, 4, 5, 7}, and μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Solution:

The given sets are A = {1, 2, 4, 5, 7}, and μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

The aim is to prove the property of the double complement of sets, which is (A')' = A.

A = {1, 2, 4, 5, 7}

A' = μ - A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {1, 2, 4, 5, 7} = {3, 6, 8, 9, 10}

(A')' = μ - A' = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {3, 6, 8, 9, 10} = {1, 2, 4, 5, 7}

The above set (A')' is the same as the given set A, and we have (A')' = A.

Therefore the given set follows the double complement property of sets.

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Practice Questions on Properties Of Sets

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FAQs on Properties Of Sets

What Are Properties Of Sets In Algebra?

The properties of sets help in performing numerous operations across sets. The commutative property, associative property, distributive property are a few properties of sets that are similar to the properties of real numbers. The properties of sets are useful to perform operations such as union, intersection, the complement of a set. The three important properties of sets are as follows.

Commutative Property:A U B = B U A
Associative Property: (A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive Property: A U (B ∩ C) = (A U B) ∩ (A U C)

How Many Properties Of Sets Are There?

There are six properties of sets. The six properties of sets are commutative property, associative property, distributive property, identity property, complement property, idempotent property. The formulas of the six properties are as follows.

Commutative Property:A U B = B U A; A ∩ B = B ∩ A
Associative Property: (A ∩ B) ∩ C = A ∩ (B ∩ C); (A U B) U C = A U (B U C)
Distributive Property: A U (B ∩ C) = (A U B) ∩ (A U C); A ∩ (B U C) = (A ∩ B) U (A ∩ C)
Identity Property: A U ∅ = A; A ∩ U = A
Complement Property: A U A' = U
Idempotent Property: A ∩ A = A; A U A = A

What Are Properties Of Union Of Sets?

The three important properties of the union of sets are:

A ∪ B = B ∪ A (Commutative law)
( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative Law)
A ∪ A = A (Idempotent law)

What Are the Properties Of Intersection Of Sets?

The properties of the intersection of sets are as listed below.

A ∩ B = B ∩ A (Commutative law).
( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
φ ∩ A = φ, U ∩ A = A (Law of φ and U).
A ∩ A = A (Idempotent law)
A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over ∪

What Are the Properties Of Compliment Of Sets?

The important properties of the complement of the set are as follows.

Complement Laws: A ∪ A′ = U (ii) A ∩ A′ = φ
De Morgan's Laws: (i). (A ∪ B)´ = A′ ∩ B′
De Morgan's Laws: (ii) (A ∩ B)′ = A′ ∪ B′
Law of Double Complementation:(A')' = A
Law of Empty Set and Universal Set: φ′ = U and U′ = φ

What Is The Use Of Properties Of Sets?

The properties of sets are useful to perform numerous operations across sets. The operations of union, intersection, complement can be conveniently performed using properties of sets.

What Are The Properties Of Sets Of Equal Sets?

The equal sets have the same number of elements, and the elements in both the sets are also the same. The cardinality of equal sets is equal.

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