Number System

Number System Formula

(a+b)^2 = a^2+b^2+2ab

(a-b)^2=a^2+b^2-2ab

(a^2-b^2)=(a+b)(a-b)

(a+b)^2-(a-b)^2=4ab

(a+b)^2+(a-b)^2=2(a^2+b^2)

(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)

(a+b)^3=a^3+b^3+3ab(a+b)

(a-b)^3=a^3+b^3+3ab(a-b)

(a^3+b^3)=(a+b)(a^2+b^2-ab)

(a^3-b^3)=(a+b)(a^2+b^2+ab)

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Set Theory

SET

A well defined collection of distinct objects is called a set. When we say, ‘well defined’, we mean that there must be given a rule or rules with the help of which we should readily be able to say that whether a particular object is a member of the set or is not a member of the set. The sets are generally denoted by capital letters A,B,C,….X,Y,Z . The members of a set are called its elements. The elements of a set are denoted by small letters a, b, c,..…,x, y, z. If an element a belong to a set A than we write a ∈ A and if a does not belong to set A then we write a ∉ A.

 LAW OF ALGEBRA OF SETS

1.            Idempotent Laws : For any set A,

(i)           A ∪ A = A

  (ii)           A ∩ A = A .

2.            Identity Laws : For any set A,

(i)           A ∪ö = A

  (ii)           A ∩U = A

i.e., ö and are identity elements for union and intersection respectively.

3.            Commutative Laws : For any two sets A and B,

                       (i)  A ∪ B = B ∪ A

                      (ii)           A ∩ B = B ∩ A

                    i.e. union and intersection are commutative.

4.            Associative Laws : If A, B and C are any three sets, then

                    (i)     ( A ∪ B) ∪ C = A ∪ ( B ∪ C )

                   i.e. union and intersection are associative.

                   (ii) A ∩ ( B ∩ C ) = ( A ∩ B) ∩ C

5.            Distributive Laws : If A, B and C are any three sets, the                                              
                     (i) A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C )

                     (ii) A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C )

                       i.e. union and intersection are distributive over intersection and union respectively.

6.            De-Morgan’s Laws : If A and B are any two sets, then

                                (i)            ( A ∪ B)′ = A′ ∩ B

                                 (ii)           ( A ∩ B)′ = A′ ∪ B′

7.            More results on operations on sets

If A and B are any two sets, then

(i)             A − B = A ∩ B′

(ii)     B − A = B ∩ A′

(iii)    A − B = A ⇔ A ∩ B = ö

(iv)    ( A − B) ∪ B = A ∪ B

(v)      ( A − B) ∩ B = ö

(vi)         A ⊆ B ⇔ B′ ⊆ A′

(vii)      ( A − B) ∪ ( B − A) = ( A ∪ B) − ( A ∩ B) .

If A, B and C are any three sets, then

 (i)A − ( B ∩ C ) = ( A − B) ∪ ( A − C )

 (ii)A − ( B ∪ C ) = ( A − B) ∩ ( A − C )

 (iii)A ∩ ( B − C ) = ( A ∩ B) − ( A ∩ C )

              (iv)A ∩ ( B ∆ C ) = ( A ∩ B) ∆ ( A ∩ C )

8.        Important results on number of elements in sets

If A, B and C are finite sets, and U be the finite universal set, then

(i)n ( A ∪ B) = n ( A) + n ( B) − n ( A ∩ B)

(ii)n ( A ∩ B) = n ( A) + n ( B) ⇔ A, B

                      are disjoint non

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