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 ‘∩.’ The intersection of ‘n’ number of sets is expressed as Set 1 ∩ Set 2 ∩ Set 3 ∩ … ∩ Set n.
For Two Sets
Formula
The formula for the intersection of two sets (A intersection B) is given by
A ∩ B = {x | x ∈ A and x ∈ B}
This means x is an element of A ∩ 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 ∩ B and read as ‘A intersection B.’
Example
If A = {3, 7} and B = {4, 7, 9} are two sets, then A ∩ 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 ∩ B = { } = ɸ Thus, the sets are disjoint.
Number of Elements in Set A ∩ 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 ∩ B) = n(A) + n(B) – n(A ∪ B)
Here,
n(A ∪ B) = Cardinality of set A ∪ B
n(A) = Cardinality of set A
n(B) = Cardinality of set B
n(A ∩ B) = Cardinality of set A ∩ B
For Three Sets
If A, B, and C are three sets, their intersection is written as A ∩ B ∩ 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 ∩ B ∩ C = {x | x ∈ A and x ∈ B and x ∈ C}
Examples
If A = {3, 7, 9}, B = {7, 9, 11, 13}, and C = {4, 7, 9} are three sets, then A ∩ B ∩ C is the element common to the sets A, B, and C.
Here, A ∩ B ∩ 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 ∩ B ∩ 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 ∩ B = {6, 8, 9, 11, 12, 14, 15} ∩ {8, 9, 11, 13, 14} = {8, 9, 11, 14} Now, (A ∩ B) ∩ C = {8, 9, 11, 14} ∩ {9, 11, 13, 15} = {9, 11} Thus, A ∩ B ∩ C = {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 ∩ B. It is represented by (A ∩ B)’ and is read as ‘A intersection B complement.’
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 ∩ B = {16}
Then, (A ∩ B)’ = U – (A ∩ 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
A ∩ B = B ∩ A
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 ∩ B = {u, v, w, x} ∩ {w, x, y, z} = {w, x}, and B ∩ A = {w, x, y, z} ∩ {u, v, w, x} = {w, x} Thus, A ∩ B = B ∩ A, verified.
Associative
If A, B, and C are three sets, then according to the associative law
(A ∩ B) ∩ C = A ∩ (B ∩ C)
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 ∩ B = {2, 4, 6, 8} ∩ {4, 6, 12} = {4, 6} B ∩ C = {4, 6, 12} ∩ {2, 4, 8} = {4} (A ∩ B) ∩ C = {4, 6} ∩ {2, 4, 8} = {4} and A ∩ (B ∩ C) = {2, 4, 6, 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
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Empty Set (ɸ)
The intersection of any set with an empty set produces the empty set.
A ∩ ɸ = ɸ
If A = {2,10, 25} and ɸ is an empty set, then A ∩ ɸ = {2, 10, 25} ∩ { } = { } = ɸ
Universal Set (U)
The intersection of any set with the universal set produces the original set.
A ∩ U = U
If A = {2,10, 25} and U = {1, 2, 5, 10, 15, 20, 25}, then A ∩ U = {2,10, 25} ∩ {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.
A ∩ A = A
Given A = {19, 38, 57}. Verify the idempotent law for the intersection.
Solution:
Given, A = {19, 38, 57} A ∩ A = {19, 38, 57} ∩ {19, 38, 57} = {19, 38, 57} = A Thus, the idempotent law for intersection is verified.
Union and Intersection of Sets
Union of Sets
Intersection of Sets
The union of sets is a new set containing all the given set elements.
The intersection of sets is a new set that contains all common elements of the given sets.
Denoted by the symbol ‘∪.’
Denoted by the symbol ‘∩.’
Written as A ∪ B and read as ‘A union B.’
Written as A ∩ B and read as ‘A intersection B.’
In set builder notation, it can be expressed asA ∪ B = {x | x ∈ A or x ∈ B}
In set builder notation, it can be expressed asA ∩ B = {x | x ∈ A and x ∈ B}
Example:If A = {3, 13} and B = {3, 7, 17}, then A ∪ B = {3, 7, 13, 17}
Example:If A = {3, 13} and B = {3, 7, 17}, then A ∩ B = {3}
