Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 The intersection of<a></a><a>sets</a>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.

2 Properties of Intersection of Sets

3 The following are some properties of the intersection operation:

4 Commutative law: A ∩ B=B ∩ A

5 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

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

7 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<a></a><a>empty set</a>.)

8 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.)

9 Hence, (A ∩ B) ∩ C=A ∩ (B ∩ C)

10 Law of φ and U:φ ∩ A=φ, U ∩ A=A

11 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

12 The intersection with the<a>universal set</a>gives back the original set A, since all its elements are in U.

13 Idempotent law: A ∩ A=A

14 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

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

16 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}

17 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}

18 Intersection of Sets Venn Diagram

19 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<a>Venn diagram</a>for the intersection of three sets.