A intersection B intersection C - (A ∩ B ∩ C)
2026-02-28 17:45 Diff
https://brightchamps.com/en-us/math/algebra/a-intersection-b-intersection-c

The intersection of sets refers to the collection of elements common to all the given sets. For example, set A students who take German and set B students who take Japanese, the intersection of the sets of students who take German and those who take Japanese includes only students who are enrolled in both language classes.

Properties of Intersection of Sets


The following are some properties of the intersection operation:

Commutative law: A ∩ B=B ∩ A


Examine the two sets A = {5, 6, 7, 8, 9, 10} and B = {6, 7, 9, 11}
Here, A ∩ B = {5, 6, 7, 8, 9, 10} ∩ {6, 7, 9, 11} = {6, 7, 9}
B ∩ A = {6, 7, 9, 11} ∩ {5, 6, 7, 8, 9, 10} = {6, 7, 9}
Hence, A ∩ B=B ∩ A
 

Associative law: (A ∩ B) ∩ C=A ∩ (B ∩ C)

Examine the three sets A = {5, 6, 7, 8}, B = {7, 8, 9, 10}, and C = {9, 10, 11, 12}.
Now,
A ∩ B={5, 6, 7, 8} ∩ {7, 8, 9, 10}={7,8}
(A ∩ B) ∩ C = {7,8} ∩ {9, 10, 11, 12}={}=φ (No elements are intersecting, so it is an empty set.)

Now, let us find A ∩ (B ∩ C)
B ∩ C={7, 8, 9, 10} ∩ {9, 10, 11, 12}={9, 10}
A ∩ (B ∩ C)= {5, 6, 7, 8} ∩ {9, 10}={}=φ (No elements are intersecting, so it is an empty set.)

Hence, (A ∩ B) ∩ C=A ∩ (B ∩ C) 

Law of φ and U: φ ∩ A=φ, U ∩ A=A

Now examine, φ = {} and A = {12, 14, 11}
φ ∩ A={} ∩ {12, 14, 11}={}=φ (The intersection with the empty set gives the empty set. 
Let U = {3, 5, 7, 9, 11, 15, 17, 19, 21, 25} and A = {5, 9, 15, 19, 25}. Then,
U ∩ A={3, 5, 7, 9, 11, 15, 17, 19, 21, 25} ∩ {5, 9, 15, 19, 25}= {5, 9, 15, 19, 25}=A

The intersection with the universal set gives back the original set A, since all its elements are in U.

Idempotent law: A ∩ A=A


If A={e, f, g, h, i} 
Therefore, A ∩ A={e, f, g, h, i} ∩ {e, f, g, h, i}={e, f, g, h, i}=A

Distributive law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Consider the following three sets: A =  {1, 3, 5, 7}, B =  {1, 2, 4, 6}, and C =  {2, 7, 4, 8}.
B ∪ C={1, 2, 4, 6} ∪ {2, 7, 4, 8}={1, 2, 4, 6, 7, 8}

Next, A ∩ (B ∪ C)={1, 3, 5, 7} ∩ {1, 2, 4, 6, 7, 8}={1, 7} 
 A ∩ B={1} and A ∩ C={7} 
(A ∩ B) ∪ (A ∩ C)={1} ∪ {7}={1,7} 

Intersection of Sets Venn Diagram


The intersection of two sets, A and B, is shown by the shaded area in the diagram above. Likewise, as an example below, we can create a Venn diagram for the intersection of three sets.