Set Identities

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Set Identities

Introduction

The methods of expressing the same set using the names of sets and set operations are known as Set Identities. Set Identities is one of the parts of Number Sets. This chapter lists out the formulas of Set Identities which are very useful in all the competitive exams to solve the questions under Quantitative Aptitude Section. Set Identities formulae can be used as an everyday online reference guide.

Formulae

Sets: A, B, C
Universal Set: I
Complement: $$A^\prime$$
Proper Subset: A $$\subset$$ B
Empty Set: $$\emptyset$$
Union of Sets: A $$\cup$$ B
Intersection of Sets: A $$\cap$$ B
Difference of Sets: A \ B

1. A $$\subset$$ I

2. A $$\subset$$ A

3. A = B if A $$\subset$$ B and B $$\subset$$ A.

4. Empty Set
$$\emptyset \subset$$ A

5. Union of Sets
C = A $$\cup$$ B = {X $$\mid$$ X $$\in$$ A or X $$\in$$ B}

6. Commutativity
A $$\cup$$ B = B $$\cup$$ A

7. Associativity
A $$\cup$$ (B $$\cup$$ C) = (A $$\cup$$ B) $$\cup$$ C

8. Intersection of Sets
C = A $$\cup$$ B = {X $$\mid$$ X $$\in$$ A or X $$\in$$ B}

9. Commutativity
A $$\cap$$ B = B $$\cap$$ A

10. Associativity
A $$\cap$$ (B $$\cap$$ C) = (A $$\cap$$ B) $$\cap$$ C

11. Distributivity
A $$\cup$$ (B $$\cap$$ C) = (A $$\cup$$ B) $$\cap$$ (A $$\cup$$ C)
A $$\cap$$ (B $$\cup$$ C) = (A $$\cap$$ B) $$\cup$$ (A $$\cap$$ C)

12. Idempotency
A $$\cap$$ A = A
A $$\cup$$ A = A

13. Domination
A $$\cap$$ $$\emptyset$$ = $$\emptyset$$
A $$\cup$$ I = I

14. Identity
A $$\cup$$ $$\emptyset$$= A
A $$\cap$$ I = A

15. Complement
$$A^\prime$$ = {X $$\in$$ I $$\mid$$ X $$\notin$$ A}

16. Complement of Intersection and Union
A $$\cup$$ $$A^\prime$$ = I
A $$\cap$$ $$A^\prime$$ = $$\emptyset$$

17. De Morgan’s Laws
$$(A \cup B)^\prime$$ = $$A^\prime$$ $$\cap$$ $$B^\prime$$
$$(A \cap B)^\prime$$ = $$A^\prime$$ $$\cup$$ $$B^\prime$$

18. Difference of Sets
C = B \ A = {X $$\mid$$ X $$\in$$ B and X $$\notin$$ A}

19. B \ A = B \ (A $$\cap$$ B)

20. B \ A = B $$\cap$$ $$A^\prime$$

21. A \ A = $$\emptyset$$

22. A \ B = A if A $$\cap$$ B = $$\emptyset$$

23. (A \ B) $$\cap$$ C = (A $$\cap$$ C)\ (B $$\cap$$ C)

24. $$A^\prime$$ = I \ A

25. Cartesian Product
C = A $$\times$$ B = {(x,y)$$\mid$$ X $$\in$$ A and Y $$\in$$ B}