Page 9 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Notes
Example 1: If A = {a, b, c} then P(A) =

{ , {a}, {b}, {c}, {a, b} {b, c}, {a, c}, {a, b, c} }.
3
The total number of these elements of power set is 8, i.e. 2 .
The sets A and B are equal if A is a subset of B and also B is a subset of A.

If U be the universal set, the set of those elements of U which are not the elements of A is
defined to be the complement of A. It is denoted by A’. Thus

A’ = {x : x U and x A}
Obviously, {A’}’ = A, ’ = U, U’ = .
It is easy to see that if A B, then A’ B’.

The difference of two sets A and B in that order is the set of elements which belong to A but
which do not belong to B. We denote the difference of A and B by A ~ B or A – B, which is read
as “A difference B” or “A minus B”. Symbolically A – B = {x : x A and x B}. It is obvious that
A – A = , and A – = A.

Union and Intersection

Let A and B be two sets. The union of A and B is the set of all elements which are in set A or in set
B. We denote the union of A and B by A B, which is usually read as “A union B”.
Symbolically, A B = {x : x A or x B}

On the other hand, the intersection of A and B is the set of all elements which are both in A and
B. We denote the intersection of A and B by A B, which is usually read as “A intersection B”.
Symbolically,
A B = {x : x A or x B}
or A B = {x : x A, x B}.

The union and intersection of sets have the following simple properties:

(i) A B B A and
A B B A Commutative laws

(ii) A (B C ) (A ) B C and Associative laws
A (B C ) (A ) B C

(iii) A A A and Idempotent laws
A A A

A (B C ) (A ) B (A C ) and
(iv) Distributive laws
A (B C ) (A ) B (A C )

(v) A (B C ) (A B ) (A C ) and
A (B C ) (A B ) (A C ) De Morgan's laws

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