Rivalry2

HTML Diff

0 added 0 removed

Original 2026-01-01

Modified 2026-02-28

1 The properties of intersection of sets are:

2 <ul><li>Commutative Law</li>

3 </ul><ul><li>Associative Law</li>

4 </ul><ul><li>Distributive Law</li>

5 </ul><ul><li>Law of Empty Set</li>

6 </ul><ul><li>Law of Universal Set</li>

7 </ul><ul><li>Idempotent Law</li>

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

9 Rule: A ∩ B = B ∩ A

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

11 A ∩ B = {cat} - (1)

12 B ∩ A = {cat} - (2)

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

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

15 Rule: (A ∩ B) ∩ C = A ∩ (B ∩ C)

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

17 A ∩ B = {2}

18 (A ∩ B) ∩ C = {2} - (1)

19 B ∩ C = {2}

20 A ∩ (B ∩ C) = {2} - (2)

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

22 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.

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

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

25 B ∪ C = {blue, green, yellow}

26 A ∩ (B ∪ C) = {blue} - (1)

27 A ∩ B = {blue}

28 A ∩ C = {blue}

29 (A ∩ B) ∪ (A ∩ C) = {blue} - (2)

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

31 Therefore, the distributive law is true.

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

33 Rule: A ∩ ∅ = ∅

34 Example: A = {1, 2, 3} and ∅ = {}

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

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

37 Rule: A ∩ U = A

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

39 A ∩ U = {sun, moon}

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

41 Idempotent Set: Intersecting a<a>set</a>with itself results in the same set.

42 Rule: A ∩ A = A

43 Example: A = {10, 20}

44 A ∩ A = {10, 20}

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