Set and notation
A well defined collection of objects
is known as a set. The set is denoted by the capital letters and its elements
by small letters.
Subset and proper subset
A set A is subset of set B, if every
elements of set A is also an element of B. It is represented by A Í B.
A set A is proper subset of set B if A
is subset of B and A ¹ B. It is denoted by A Ì B and
set A is said to proper subset of B and set B is said to be super set of A.
Universal set
A fixed set such all the sets under consideration
are the subsets of that fixed set, then the fixed set is called the universal
set. The universal set is denoted by U.
Empty or Null set
A set having no element is called empty or null set. It is
denoted by f or { }.
Finite set and infinite set
A set having a finite number of
elements is known as a finite set. The set which consists infinite number of
elements is called an infinite set.
Equal and Equivalent sets
Two sets A and B are said to be equal
if they have the same elements. If A and B are equal then, we write A = B.
Thus, if A Í B and B Í A, then A = B.
Two sets A and B are said to be
equivalent if they have the same number of elements. The equivalent sets A and
B are denoted by A~B.
Intersecting and disjoint sets
If two sets A and B have at least one
element in common, then A and B are called intersecting sets. If sets A and B
have no elements in common, then these sets are called disjoint sets.
Power set
The collection of all possible subsets
of any set S is called the power set of S. The power set of S is denoted by 2s.
If S = {a, b}, then 2S = {f, {a}, {b}, {a, b}}.
Venn diagram
The diagrammatic representation of
sets, relation of sets and operation on sets is known as Venn diagram. It
consists of a universal set U represented by a rectangle, subset of U by the
closed curve and the elements of set by the points within the closed curve.
Operation on sets:
· Union of two sets: The union of the sets A and B is defined as the set of all elements which
belong to A or B or both.
In symbol, A È B = {x : x ÎA or x Î B}.
· Intersection of two sets: The intersection of sets A and B is defined as the set of
all elements which belong to both sets A and B.
In symbol, A Ç B = {x:
xÎ A and x ÎB}
· Difference of two sets: The difference of set B from set A is denoted by A - B is a
set which consists all elements of set A but not belonging to set B.
In symbol, A - B = {x: x Î A and x Ï B}
· Complement of set: The complement set A is denoted by AC is a set all the
elements of universal set U that do not belong to set A.
In symbol, AC
= {x: x Î U and x Ï A}
·
Symmetric difference: The symmetric difference of set A and
B is denoted by A D B is the set which is the union of
the difference A - B and B - A.
In symbol, A D B = (A - B) È (B - A)
Cardinal number of a finite set
The number of distinct elements of a
finite set is called the cardinal number of the set. The cardinal number of a
finite set A is denoted by n(A).
Some Importance formulae
· A È A = A, A Ç A = A
(Idempotent law)
· A È B = B È A, A Ç B = B Ç A (Commutative law)
· (A È B) È C = A È (B È C), (A Ç B) Ç C = A Ç (B Ç C) (Associative law)
· A È (B Ç C) = (A È B) Ç (A È C), A Ç (B È C) = (A Ç B) È (A Ç C)
(Distributive law)
· A È U = U, A Ç U = A, A
È f = A,
A Ç f = f (Identity law)
· A È A' = U, A Ç A' = f, (A')' =
A, U' = f, f' = U (Complement law)
· (A È B)' = A' Ç B', (A Ç B)' = A'
È B' (De Morgan's law)
· n(A È B) = n(A) + n(B) - n(A Ç B)
· n(A È B È C) = n(A) + n(B) + n(C) - n(A Ç B) - n(B Ç C) - n(C Ç A) + n(A
Ç B Ç C)
· n(AÈB) + n(AÇB)C = n(U)
Real number system
Natural numbers
The set of numbers which are used for
counting is called natural numbers. In other words, the set of natural numbers
consists 1 and each natural numbers has a successor n + 1. It is denoted by N
or Z+.
Integers
A number is said to be a member of set
of integers, if it consists positive natural number and negative of natural
numbers with zero. It is denoted by Z or I.
Rational numbers
A number is said to a rational number
if it can be expressed as m/n where n ¹ 0 and m, n Î Z. A set
of rational number is denoted by Q.
Irrational numbers
A real number which is not rational
(it cannot be expressed as m/n) is called an irrational number. A set of
irrational number is denoted by R - Q or Ir, where Q denotes
the set of rational numbers and R denotes as universal set of real numbers.
Real numbers
A set of numbers which consists of all
set of rational and the set of irrational numbers is called the set of real
numbers. It is denoted by R.
Interval
Let a and b be two numbers on the real
line. Then the set of points on the real line between a and b is known as an
interval. The real numbers a and b are known as the end points of the interval.
An interval not containing the end points a and b is known as open interval.
The open interval is denoted by (a, b).
In symbol, (a, b) = {x: a < x < b}
An interval containing both the end points a and b is known as the closed interval and it is denoted by [a, b].
In symbol, [a, b] = {x: a £ x £ b}.
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