JEE Main Maths Formulae

FORMULAE

Properties:

1. A ∪ B = B ∪ A
2. (A∪ B) ∪ C = A ∪(B ∪ C)
3. A ∪ ⌀ = A
4. A ∪ A = A
5. A ∪ A’ = U
6. If A ⊂ B then A ∪ B
7. U ∪ A = U
8. A ⊂ (A ∪ B),B ⊂ (A ∪ B)
9. A ∩ B = B ∩ A
10. ( A ∩B ) ⊂ A, (A ∩ B) ⊂ B
11. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
12. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
13. (A ∩ B) ∩ C = A ∩ (B ∩ C)
14. ⌀ ∩ A = ⌀
15. A ∩ A = A
16. A ∩ A’ = ⌀
17. If A ⊂ B then A ∩ B = A
18. U ∩ A =A
19. (A’)’ = A
20. ⌀’ = U where U is the universal set
21. U’ = ⌀

Difference of sets:

1. A – B is a subset of A and B – A is a subset of B.
2. The sets A – B, A ∩ B,B-A are mutually disjoint sets. The intersection of any of these two sets is the null set.
3. A – B = A ∩ B'
4. B – A = A’ ∩ B
5. A ∪ B=(A-B) ∪ (A ∩ B)∪(B-A)
   
   If n(A) = m then n[P(A)] = 2^m where n[P(A)] is the number of subsets of set A.
   If n(A) = m then n[P(A)] = 2^m – 1 where n[P(A)] is the number of proper subsets of set A.

De Morgan’s Laws:
For any two sets A and B

1. (A ∪ B)'=A' ∩ B'
2. (A ∩ B)'=A' ∪ B'

Some Important Results:
For given sets A, B

1. n(A ∪ B)=n(A)+ n (B) - n(A ∩B)
2. When A and B are disjoint sets, then n(A ∪ B)=n (A)+ n(B)
3. n(A ∩ B')+ n (A ∩ B)= n(A)
4. n(A’ ∩ B)+ n (A ∩ B)= n(B)
5. n(A ∩ B')+ n(A ∩ B)+ n(A’ ∩B)= n(A ∪ B)
   For any sets A, B, and C
6. n(A ∪ B ∪ C)=n(A)+ n(B)+ n(C)-n(A ∩B)-n(B∩C)-n(C∩A)+n(A∩B∩C)

Important results on Cartesian Product of sets

1. A × (B ∪ C)=(A×B) ∪(A×C)
2. A × (B ∩ C)=(A×B) ∩(A×C)
3. A × (B - C)=(A×B)-(A×C)
4. If A and B are any two non-empty sets and A ×B=B ×A then A=B
5. If A ⊆ B,then A×A ⊆(A×B)∩ (B ×A)
6. If A ⊆ B,then A×C ⊆(B×C) for any set C.
7. If A ⊆ B and C ⊆ D then A ×C ⊆ B×D
8. For any set A, B, C, D
   (A × B)∩(C × D)=(A ∩ C)×(B ∩ D)
9. If A and B are two non-empty sets and A ∩ B has n elements,then A×B and B×A have n² elements in common.

Various types of relations :
Let A be a non-empty set then a relation R on A is said to be

1. Reflexive if (a, a) ∈ R for all a ∈A i. e. aRa for all a ∈A
2. Symmetric if (a, b) ∈ R then (b, a) ∈ R for all a, b ∈ A i. e. aRb then bRa for all a, b ∈ A
3. Transitive if (a, b) ∈ R and (b, c) ∈R  then (a,c)∈R for all a, b, c ∈A
   i.e. aRb, bRc then aRc for all a, b, c ∈ A
4. A relation which is reflexive, symmetric and transitive is called an equivalence relation.

Types of functions:

1. One-one function :
   A function A → B is said to be one-one if distinct elements in A have distinct images in B. i.e. x₁, x₂ ∈A such that x₁ ≠ x₂ then f(x₁) ≠ f(x₂)
2. Onto Function :
   A function A → B is said to be onto if every element of B is the image of some element of A under the function f. If f is onto then for every y ∈ B their exists at least one element x ∈ A such that y=f(x). If f is onto then range of f = codomain of f
3. Into function :
   The function A → B is such that there exists at least one element in B which has no preimage in A, then f is said to be an into function.
   Hence, the range of f is a proper subset of its codomain B.
4. Even and odd function :
   A function f is said to be an even function, if f(-x) = f(x) for all x ∈R
   A function f is said to be an odd function, if f(-x) = -f(x) for all x ∈R
5. The greatest integer function (or step function) :
   If x ∈R then the greatest integer not exceeding x is denoted by [x].
6. Composite function :
   Let f:A → B and g:B → C be two functions where domain of g is same as co-domain of f then we define the composite function
   of f and g from A to C denoted by g^∘f(x) = g[f(x)]
7. Inverse function :
   If f:A → B is one-one onto, g:B → A which is also one-one, onto such that g^∘f:A →A and
   f∘g:B →B are both identity functions then f and g are called as inverse functions of each other. Function g is denoted by f^-1 and is read as f-inverse. Hence, a function f^-1 as f^-1 : B →A such that if f(x)= y then f^-1 (y) = x.