sets imp points

SETS
CONCEPTS AND RESULTS
** Set : a set is a well-defined collection of objects.
If a is an element of a set A, we say that “ a belongs to A” the Greek symbol  (epsilon) is used
to denote the phrase „belongs to‟. Thus, we write a  A. If „b‟ is not an element of a set A, we write b
 A and read “b does not belong to A”.
There are two methods of representing a set :
(i) Roster or tabular form (ii) Set-builder form.
In roster form, all the elements of a set are listed, the elements are being separated by commas
and are enclosed within brackets { }. For example, the set of all even positive integers less than 7 is
described in roster form as {2, 4, 6}.
In set-builder form, all the elements of a set possess a single common property which is not
possessed by any element outside the set. For example, in the set {a, e, i, o, u}, all the elements possess
a common property, namely, each of them is a vowel in the English alphabet, and no other letter
possess this property. Denoting this set by V, we write V = {x : x is a vowel in English alphabet}
** Empty Set : A set which does not contain any element is called the empty set or the null set or the
void set. The empty set is denoted by the symbol φ or { }.
** Finite and Infinite Sets : A set which is empty or consists of a definite number of elements is
called finite otherwise, the set is called infinite.
** Equal Sets : Two sets A and B are said to be equal if they have exactly the same elements and we
write
A = B. Otherwise, the sets are said to be unequal and we write A ≠ B.
** Subsets : A set A is said to be a subset of a set B if every element of A is also an element of B.
In other words, A  B if whenever a  A, then a  B. Thus A  B if a  A  a  B
If A is not a subset of B, we write A  B.
** Every set A is a subset of itself, i.e., A  A.
** φ is a subset of every set.
** If A  B and A ≠ B , then A is called a proper subset of B and B is called superset of A.
** If a set A has only one element, we call it a singleton set. Thus, { a } is a singleton set.
** Closed Interval : [a , b] = {x : a ≤ x ≤ b}
** Open Interval : (a , b) = { x : a < x < b}
** Closed open Interval : [a , b) = {x : a ≤ x < b}
** Open closed Interval : (a , b] = { x : a < x ≤ b }
** Power Set : The collection of all subsets of a set A is called the power set of A. It is denoted by
P(A)
If A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2m.
** Universal Set : The largest set under consideration is called Universal set.
** Union of sets : The union of two sets A and B is the set C which
consists of all those elements which are either in A or in
B (including those which are in both). In symbols, we write.
A  B = { x : x  A or x  B }.
x  A  B  x  A or x  B
x  A  B  x  A and x  B
** Some Properties of the Operation of Union
(i) A  B = B  A (Commutative law)
(ii) ( A  B )  C = A  ( B  C) (Associative law )
(iii) A  φ = A (Law of identity element, φ is the identity of  )

, (iv) A  A = A (Idempotent law)
(v) U  A = U (Law of U)
** Intersection of sets : The intersection of two sets A and B is the
set of all those elements which belong to bothA and B.
Symbolically, we write A ∩ B = {x : x  A and x  B}
x  A ∩ B  x  A and x  B
x  A ∩ B  x  A or x  B
** Disjoint sets : If A and B are two sets such that A ∩ B = φ, then
A and B are called disjoint sets.


** Some Properties of Operation of Intersection
(i) A ∩ B = B ∩ A (Commutative law).
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
(iii) φ ∩ A = φ, U ∩ A = A (Law of φ and U).
(iv) A ∩ A = A (Idempotent law)
(v) A ∩ ( B  C ) = ( A ∩ B )  ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over 
** Difference of sets : The difference of the sets A and B in this order
is the set of elements which belong to A but not to B.
Symbolically, we write A – B and read as “ A minus B”.
A – B = { x : x  A and x  B }.


* The sets A – B, A ∩B and B – A are mutually disjoint sets,
i.e., the intersection of any of these two sets is the null set.




** Complement of a Set : Let U be the universal set and A a subset of U.
Then the complement of A is the set of all elements of U which
are not the elements of A. Symbolically, we write A′ to denote
the complement of A with respect to U.
Thus, A′ = {x : x  U and x  A }. Obviously A′ = U – A
** Some Properties of Complement Sets
1. Complement laws: (i) A  A′ = U (ii) A ∩ A′ = φ
2. De Morgan‟s law: (i) (A  B)´ = A′ ∩ B′ (ii) (A ∩ B )′ = A′  B′
3. Law of double complementation : (A′ )′ = A
4. Laws of empty set and universal set φ′ = U and U′ = φ.

** Practical Problems on Union and Intersection of Two Sets :
(i) n ( A  B ) = n ( A ) + n ( B ) – n ( A ∩ B )
(ii) n ( A  B ) = n ( A ) + n ( B ) , if A ∩ B = φ.
(iii) n (A  B  C ) = n ( A ) + n ( B ) + n (C) – n (A ∩ B) – n (B ∩ C) – n (A ∩ C) + n (A ∩ B ∩ C).