Here we will learn about some of the laws of algebra of sets.
1. Commutative Laws:
For any two finite sets A and B;
(i) A U B = B U A
(ii) A ∩ B = B ∩ A
2. Associative Laws:
For any three finite sets A, B and C;
(i) (A U B) U C = A U (B U C)
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)
Thus, union and intersection are associative.
3. Idempotent Laws:
For any finite set A;
(i) A U A = A
(ii) A ∩ A = A
4. Distributive Laws:
For any three finite sets A, B and C;
(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)
Thus, union and intersection are distributive over intersection and union respectively.
5. De Morgan’s Laws:
For any two finite sets A and B;
(i) A – (B U C) = (A – B) ∩ (A – C)
(ii) A - (B ∩ C) = (A – B) U (A – C)
De Morgan’s Laws can also we written as:
(i) (A U B)’ = A' ∩ B'
(ii) (A ∩ B)’ = A' U B'
More laws of algebra of sets:
6. For any two finite sets A and B;
(i) A – B = A ∩ B'
(ii) B – A = B ∩ A'
(iii) A – B = A ⇔ A ∩ B = ∅
(iv) (A – B) U B = A U B
(v) (A – B) ∩ B = ∅
(vi) A ⊆ B ⇔ B' ⊆ A'
(vii) (A – B) U (B – A) = (A U B) – (A ∩ B)
7. For any three finite sets A, B and C;
(i) A – (B ∩ C) = (A – B) U (A – C)
(ii) A – (B U C) = (A – B) ∩ (A – C)
(iii) A ∩ (B - C) = (A ∩ B) - (A ∩ C)
(iv) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)
● Set Theory
● Sets
● Subset
● Practice Test on Sets and Subsets
● Problems on Operation on Sets
● Practice Test on Operations on Sets
● Venn Diagrams in Different Situations
● Relationship in Sets using Venn Diagram
● Practice Test on Venn Diagrams
8th Grade Math Practice
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