Definitions 
1. Set: A well defined collection of distinct objects is called a set. 

2. Representation of Sets: (i) Roaster or Tabular form (ii) Set builder form

3. Empty set: A set which does not contain any element is called an empty set or null set or void set. It is denoted by ϕ or {}.

4. Singleton set: A set consisting of a single element is called a singleton set.

5. Finite set: A set which is empty or consists of a definite/limited/countable number of elements is called a finite set.

6. Infinite set: A set which is not finite or consists of unlimited number of elements is called an infinite set.

7. Equivalent sets: Two finite sets A and B are equivalent if their cardinal numbers are same i.e. n(A)=n(B)

8. Equal sets: Two sets are said to be equal if they have exactly the same elements.

9. Subset: If every element of A is an element of B, then then A is called a subset of B i.e. A⊂B

10. Proper subset: If A⊂B and A≠B, then A is called a proper subset of B. 

11. Universal set: If all the sets under consideration are subsets of a large set U, then U is known as a universal set.

12. Power set: The collection of all subsets of a set A is called power set and is denoted by P(A). 

13. Venn-Diagram: A geometrical figure illustrating universal set, subsets and their operations is known as Venn-Diagram.

14. Union of sets: The union of two sets A and B is the set containing all the elements of A and B, denoted by A∪B.

15. Intersection of sets: The intersection of two sets A and B is the set containing all the common elements of A and B, denoted by A∩B.

16. Disjoint sets: two sets A and B are said to be disjoint, if A∩B=ϕ

17. Difference of sets: Difference of two sets i.e. (A-B) is the set containing all elements of A which do not belong to B.

18. Symmetric difference: Symmetric difference of  two sets A and B is, A∆B=(A-B)∪(B-A)

19. Complement of a set: A'=U-A

Algebra of Sets

1. Idempotent Laws
(i) A∪A = A
(ii) A∩A = A

2.Identity Laws
(i) A∪ϕ = A
(ii) A∩U = A

3. Boundedness Laws
(i) A∪U = U
(ii) A∩ϕ = ϕ

4. Laws of complementation
(i) U' = ϕ
(ii) ϕ' = U
(iii) (A')' =A
(iv) A∪A' = U
(v) A∩A' = ϕ

5. Commutative Laws
(i) A∪B = B∪A
(ii) A∩B = B∩A

6. Associative Laws
(i) (A∪B)∪C = A∪(B∪C)
(ii)  (A∩B)∩C = A∩(B∩C)

7. Distributive Laws
(i) A∪(B∩C) = (A∪B)∩(A∪C)
(ii) A∩(B∪C) = (A∩B)∪(A∩C)

8. De Morgan's Laws
(i) (A∪B)' = A' ∩ B'
(ii) (A∩B)' = A' ∪ B'

9.(i) If x∈(A∪B) ⇔ x∈A or x∈B
(ii) If x∉(A∪B) ⇔ x∉A and x∉B
(iii) If x∈(A∩B) ⇔ x∈A and x∈B
(iv) If x∉(A∩B) ⇔ x∉A or x∉B

10. If A⊂B then
(i) A∪B = B
(ii) A∩B = A
(iii) A−B = ϕ
(iv) B−A may or may not be null

11. If A= ϕ then
(i) A∪B = B
(ii) A∩B = ϕ
(iii) A−B = ϕ
(iv) B−A= B

12. If A=B then
(i) A−B = ϕ
(ii) B−A = ϕ

13. If A⊂B and B⊂A ⇔ A = B

14. If A⊂B and B⊂C, then A⊂C

15. (i) A⊂(A∪B)
(ii) B⊂(A∪B)
(iii) (A∩B)⊂A
(iv) (A∩B)⊂B

16. If (A∩B) = ϕ then
(i) A−B = A
(ii) B−A = B

17. (i) A-B = A∩B'
(ii) B-A = B∩A'

18. (A−B)∩(B−A) = ϕ

19. (i) A∪(A∩B) = A
(ii) A∩(A∪B) = A

20. (i) A−(B∪C) = (A−B)∩(A−C)
(ii) A−(B∩C) = (A−B)∪(A−C)

Cardinal Numbers and Properties

1. If A is set with n(A) = m, then
(i) number of subsets of A = $\displaystyle \small 2^{m}$
(ii) number of proper subsets of A = $\displaystyle \small 2^{m}$−1
(iii) n(P(A)) = $\displaystyle \small 2^{m}$

2. n(A∪B) = n(A) + n(B) [If A and B are disjoint sets]

3. n(A∪B) = n(A) + n(B) - n(A∩B)

4. n(A∪B) = n(A-B) + n(A∩B) + n(B-A)

5. n(A) = n(A-B) + n(A∩B)

6. n(B) = n(B-A) + n(A∩B)

7. n(A∪B∪C) =  n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C)

8. n(A') = n(U) - n(A) [provided U is finite]

9. n(A'∪B') = n((A∩B)') = n(U) - n(A∩B)

10. n(A'∩B') = n((A∪B)') = n(U) - n(A∪B)

*To solve problems

1. If asked to prove any conditions, prove it by taking examples.

2. If asked to prove any conditions using properties of sets (if mentioned in question), then use any laws or properties to prove.

3. If asked to show any two sets are equal, say D=E, then try to show that D⊂E and E⊂D.
Example: To show D⊂E, consider x∈D and prove x∈E.
Similarly, to show E⊂D, consider y∈E and prove y∈D.

4. If asked to show any two conditions are equal, say con1=con2, then use properties or try to show that con1⊂con2 and con2⊂con1.
Example: Show that A∪(A∩B)=A
To solve this use properties of sets or show that A∪(A∩B)⊂A and A⊂A∪(A∩B)

5. To solve problems on cardinal properties of sets, use cardinal formulas or use Venn-Diagram.
(i) A and B are two sets, then
Total= n(U)
Only in A= n(A)
Only in B= n(B)
In A but not B= n(A-B)
In B but not A= n(B-A)
Either A or B = n(A∪B)
At least in one = n(A∪B)
In both A and B= n(A∩B)
Neither A nor B= n(A'∩B')

(ii) A, B and C are sets, then
Total= n(U)
Only in A= n(A)
Only in B= n(B)
Only in C= n(C)
Either A or B or C = n(A∪B∪C)
In all A and B and C= n(A∩B∩C)
In at least one= n(A∪B∪C)
In at least two= n(A∩B)+ n(A∩C)+ n(B∩C)- 2*n(A∩B∩C)
In exactly three = n(A∩B∩C)
In exactly two = n(A∩B)+ n(A∩C)+ n(B∩C)- 3*n(A∩B∩C)
In exactly one =  n(A∪B∪C)- n(A∩B)- n(A∩C)- n(B∩C)+ 2*n(A∩B∩C)
(iii) A, B and C are sets, then using Venn-Diagram
Total= n(U)
Only in A= n(A) = a+x+y+w
Only in B= n(B) = b+x+z+w
Only in C= n(C) = c+z+y+w
Either A or B or C = n(A∪B∪C) = a+b+c+x+y+z+w
In all A and B and C= n(A∩B∩C) = w
In at least one = a+b+c+x+y+z+w
In at least two =x+y+z+w
In exactly three = w
In exactly two =x+y+z+w
In exactly one =a+b+c
In A and B= n(A∩B) = x+w
In A and C= n(A∩C) = y+w
In B and C= n(B∩C) = z+w