Intersection of Sets

The intersection of two or more sets is a set that contains all the elements common to the original sets.

Symbol

In set theory, the intersection of sets is denoted by the symbol â€˜âˆ©.â€™ The intersection of â€˜nâ€™ number of sets is expressed as Set 1 âˆ© Set 2 âˆ© Set 3 âˆ© â€¦ âˆ© Set n.

For Two Sets

Formula

The formula for the intersection of two sets (A intersection B) is given by

A âˆ© B = {x | x âˆˆ A and x âˆˆ B}

This means x is an element of A âˆ© B if and only if x is an element of A and x is an element of B.

If A and B are two sets, then the intersection of A and B is written as A âˆ© B and read as â€˜A intersection B.â€™

Example

If A = {3, 7} and B = {4, 7, 9} are two sets, then A âˆ© B = {7}, which is the only element common to sets A and B.

Find the intersection of sets A and B when A = {3, 5, 7, 9, 11} and B = {4, 8, 12, 16}

Solution:

Given, A = {3, 5, 7, 9, 11} and B = {4, 8, 12, 16}
Since there are no common elements between the two given sets
A âˆ© B = { } = É¸
Thus, the sets are disjoint.

Number of Elements in Set A âˆ© B

If A, B, and C are three sets, then the formula to find the cardinality of the intersection of two sets is given by:

n(A âˆ© B) = n(A) + n(B) – n(A âˆª B)

Here,

n(A âˆª B) = Cardinality of set  A âˆª B

n(A) = Cardinality of set A

n(B) = Cardinality of set B

For Three Sets

If A, B, and C are three sets, their intersection is written as A âˆ© B âˆ© C. It is the set whose all elements are common to A, B, and C.

Formula

The formula for the intersection of three sets (A intersection B intersection C) is given by

A âˆ© B âˆ© C = {x | x âˆˆ A and x âˆˆ B and x âˆˆ C}

Examples

If A = {3, 7, 9}, B = {7, 9, 11, 13}, and C = {4, 7, 9} are three sets, then A âˆ© B âˆ© C is the element common to the sets A, B, and C.

Here, A âˆ© B âˆ© C = {7, 9}, which are the two elements common to sets A, B, and C

Given: A = {6, 8, 9, 11, 12, 14, 15}, B = {8, 9, 11, 13, 14}, and C = {9, 11, 13, 15}, then find A âˆ© B âˆ© C

Solution:

Given, A = {6, 8, 9, 11, 12, 14, 15}, B = {8, 9, 11, 13, 14}, and C = {9, 11, 13, 15}
Now,
A âˆ© B = {6, 8, 9, 11, 12, 14, 15} âˆ© {8, 9, 11, 13, 14} = {8, 9, 11, 14}
Now, (A âˆ© B) âˆ© C = {8, 9, 11, 14} âˆ© {9, 11, 13, 15} = {9, 11}

Intersection of Sets and Venn Diagram

In the Venn diagram, the shaded region illustrates the intersection of sets A and B. It covers all the elements common to both sets.

Similarly, a Venn diagram can be drawn for the intersection of three sets: A, B, and C.

Complement of Intersection of Sets

The complement of the intersection of sets is a set containing all the elements in the universal set but absent in A âˆ© B. It is represented by (A âˆ© B)’ and is read as â€˜A intersection B complement.â€™

In the Venn diagrams below, the shaded regions mark the complement of the intersection of three sets: A, B, and C.

For example,

If A = {8, 16, 25}, B = {10, 12, 16}, and U = {6, 8, 10, 12, 16, 20, 25}, then A âˆ© B = {16}

Then, (A âˆ© B)’ = U – (A âˆ© B) = {6, 8, 10, 12, 16, 20, 25} – {16} = {6, 8, 10, 12, 20, 25}

Properties

Commutative

If A and B are two sets, then according to the commutative law

Verify the commutative law for the intersection of two sets, A and B, when A = {u, v, w, x} and B = {w, x, y, z}

Solution:

Here, A = {u, v, w, x} and B = {w, x, y, z}
Now, A âˆ© B = {u, v, w, x} âˆ© {w, x, y, z} = {w, x}, andÂ
B âˆ© A = {w, x, y, z} âˆ© {u, v, w, x} = {w, x}

Associative

If A, B, and C are three sets, then according to the associative law

Verify the associative law for the intersection of three sets, A, B, and C. Given A = {2, 4, 6, 8}, B = {4, 6, 12}, and C = {2, 4, 8}

Solution:

Given, A = {2, 4, 6, 8}, B = {4, 6, 12}, and C = {2, 4, 8}
Now, A âˆ© B = {2, 4, 6, 8} âˆ© {4, 6, 12} = {4, 6}
B âˆ© C = {4, 6, 12} âˆ© {2, 4, 8} = {4}
Thus, the associative law is verified.

Distributive

If A, B, and C are three sets, then according to the distributive law for the intersection of sets

Empty Set (É¸)

The intersection of any set with an empty set produces the empty set.

If A  = {2,10, 25} and É¸ is an empty set, then A âˆ© É¸ = {2, 10, 25} âˆ© { } = { } = É¸

Universal Set (U)

The intersection of any set with the universal set produces the original set.

If A  = {2,10, 25} and U = {1, 2, 5, 10, 15, 20, 25}, then A âˆ© U = {2,10, 25} âˆ© {1, 2, 5, 10, 15, 20, 25} = {1, 2, 5, 10, 15, 20, 25} = U

Idempotent

It states that the intersection of a set with itself results in the same original set.