Intersection of Sets
2026-02-28 00:48 Diff
https://brightchamps.com/en-us/math/algebra/intersection-of-sets

The properties of intersection of sets are:

  • Commutative Law
  • Associative Law
  • Distributive Law
  • Law of Empty Set
  • Law of Universal Set
  • Idempotent Law

Commutative Law: Commutative law states that the order of the intersection doesn’t matter. The order of the sets does not affect the result.

Rule: A ∩ B = B ∩ A


Example: Let A = {cat, dog} and B = {cat, rabbit}. 


A ∩ B = {cat} — (1)


B ∩ A = {cat} — (2)


From (1) and (2), we get that A ∩ B = B ∩ A

Associative Law: While finding the intersection of three sets, we can group any two sets first. If the resultant is grouped with the remaining set, the result will be the same. 


Rule: (A ∩ B) ∩ C = A ∩ (B ∩ C)


Example: Let A = {1, 2}, B = {2, 3}, C = {2, 4}


A ∩ B = {2}


(A ∩ B) ∩ C = {2} — (1)


B ∩ C = {2}


A ∩ (B ∩ C) = {2} — (2)


From (1) and (2), (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Law: If we intersect one set with the union of two other sets, it is equivalent to finding the intersection with each set separately and then taking their union.

 
Rule: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)


Example: A = {red, blue}, B ={blue, green}, C = {blue, yellow}


B ∪ C = {blue, green, yellow}


A ∩ (B ∪ C) = {blue} — (1)


A ∩ B = {blue}


A ∩ C = {blue}


(A ∩ B) ∪ (A ∩ C) = {blue} — (2)


From (1) and (2), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)


Therefore, the distributive law is true.

Law of Empty Set: If we take the intersection of any set with an empty set, the result will always be an empty set. 


Rule: A ∩ ∅ = ∅


Example: A = {1, 2, 3} and ∅ = {}


A ∩ ∅ = ∅, because the empty set has no elements to share with set A.

Law of Universal Set: If we intersect any set with a universal set, the result is the original set, because the universal set has all the elements in it. 

Rule: A ∩ U = A


Example: A = {sun, moon}, U = {sun, moon, stars, sky}


A ∩ U = {sun, moon}


The intersection of set A with the universal set U gives back set A. 


Idempotent Set: Intersecting a set with itself results in the same set.


Rule: A ∩ A = A


Example: A = {10, 20}


A ∩ A = {10, 20}


Intersecting a set with itself results in the same set, therefore the idempotent law is true.