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Study About Complement of Sets and Their Intersection Here

In mathematics, we perform operations such as addition, subtraction, multiplication, and others on two or more numbers to obtain a resultant value. Similarly, in set theory, certain operations are performed on two or more sets to obtain a combination of elements, depending on the operation. There are four major types of operations on sets in set theory, such as:

On that note, let’s discuss these operations two at a time thoroughly, here Intersection and Complement of sets, for a better understanding. 

Intersection of Sets 

The set that contains all common elements from both sets A and B is called a set intersection. The phrase “AND” is used to indicate the Intersection of Sets, implying that the elements in the intersection are present in both A and B. The Intersection of the set is represented by the symbol “∩”. If x ∈A ∩B, then x∈A and x∈B, according to the standard definition.

Consider two friends throwing a party and only inviting their mutual pals. They typed their friends’ names into a spreadsheet, then searched for mutual friends and invited only those; this is known as the Intersection of the set of friends.

Intersection of Sets Properties

The Intersection Operation has the following properties:

  • Commutative law: The intersection of two or more sets is governed by the commutative law, which states that if we have two sets A and B, then

A ∩ B = B ∩ A

  • Associative law: The intersection operation is governed by the associative law, which states that if we have three sets A, B, and C, then

 A ∩ (B ∩ C)= (A ∩ B) ∩ C

  • Law of φ and U: In intersection, we get all common elements between two sets. There will be no common elements between φ and A since φ has no elements.

Therefore, the intersection of φ and A is φ.

φ ∩ A = φ

Similarly, the common elements between U and A will be all elements of A since U has all the elements.

Therefore, A is the intersection of U and A.

U ∩ A = A

  • Distributive law: 

A ∩ (B U C) = (A ∩ B) U (A ∩ C), i. e., ∩ distributes over U.

Complement of a Set

If “U” is a universal set and “A” is any subset of “U,” the complement of “A” is the set of all elements of “U” which are not elements of “A.”

A’ = {x:x ∈U and x∉A}.

Complement of a Set Examples

Consider a universal set “U” of all natural numbers less than or equal to 20 to make it more clear.

Let us define set A, which is a subset of U, as the set containing all prime numbers.

Thus we can see that,

A = {2,3,5,7,11,13,17,19}

All of the elements that are present in the universal set U but not in A make up the complement of this set A.

Therefore, A’ is given by:

A’={1,4,6,8,9,10,12,14,15,16,18,20}

Properties of Complement of Set

The complement of a set has three important properties. Let’s go over the three properties of a set’s complement:

Complement Laws: This property states that the union of two sets, A and A’, yields the universal set U, of which A and A’ are subsets. 

A ∪ A’ equals U 

In addition, the intersection of a set A and its counterpart A’ yields the empty set denoted by∅.

A ∩ A’ = ∅

Law of Double Complementation: This is the second of three properties of a set’s complement. If we take the complement of the complemented set named A’, we get the set A itself, according to the law of Double Complementation.

(A’ )’ equals A

Law of Universal Set and Empty Set: According to this law, the complement of the universal set gives us the empty set and vice versa.,

∅’ equals U And U equals ∅’.

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