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Mathematics-SETS and Types of SETS
The Basic idea of a set was first introduced by Georg Cantor. A set is a well defined collection of distinct objects. Thus a set is determined, Whether the given object is belongs to or does not belong to the set. Let us show some example.
(1) Even natural numbers less than 20, i.e., 2, 4, 6, 8, 10, 12, 14, 16, 18.
(2) The districts in Kerala.
In each of the above collections we can say that given object is belongs to or does not belong to particular set. So we can say that district "Trivandrum" belongs to the collections of districts in Kerala. Just look at the below example, for identifying whether the given collection is well defined or not.
(1) The collection of best horror novels in English Language.
In the above case the collection may vary from person to person. So we can not say it is well defined.
The objects which constitute the given set, are called elements. The elements of a given set are also known as members or points of a set.
Capital Letters A, B, C............., Y, Z are used to denote sets and small letters a, b, c, ....., y, z are used to denote elements of the set. If it is an element of the set S , then we can write it in symbol as "a Є S" and read it as " a is an element of S " or " a is a member of a S " or " a belongs to S ".
For specifying a set we choose two methods.
(1) Tabular or Roster Form :: -
In this form the set is represented by listing all its elements, separating the elements by commas and enclosing them in braces.
Eg::) Z = { 0, 2, 4, 6, 8} is a set of even numbers less than 10.
(2) Rule or Set Builder Form::-
In this form members of the set are represented by stating their characteristic property.
Eg::) Z = {x|x is an even number ≤ 10 } is the set of even numbers less than or equal to 10. Here " | " or " : " is used to mean "such that".
Types of Set::-
(1) The EMPTY Set -
A set which has no element is called the empty set ( null set or void set ) denoted by ø . Here we give some examples of empty set.
Eg::) If A = { x: 8 < x < 9, x is a natural number between 8 and 9 }, then A is an empty set, because there is no natural numbers between 8 and 9 .
(2) FINITE and INFINITE Sets -
A set consisting of a definite number of elements is called a finite set.
Eg::) Let P = {2, 3, 4, 5, 6} . Then Z is a finite set of natural numbers between 1 and 7 containing 5 elements.
A set having an infinite number of elements is termed an infinite set.
Eg::) Let P = {0, 1, 2, 3, 4, ..........} . Then Z is a infinite set of whole numbers.
The following symbols are used to denote the infinite sets of real numbers .
(1) N : the set of natural numbers.
(2) Z : the set of integers.
(3) Q : the set of rational numbers.
(4) R : the set of real numbers.
(5) Z+ : the set of positive integers.
(6) Q+ : set of positive rational numbers.
(7) R+ : the set of positive real numbers.
(3) EQUAL and EQUIVALENT SETS -
Equal Sets : - Two sets A and B are said to be equal , if they consists of exactly the same elements, and we say that A = B, otherwise that sets are said to be unequal.
Eg::) A = { x|x is a letter in the word ARYA }
B = { x|x is a letter in the word RAYA } are equal sets.
Equivalent sets : - The sets which have the same number of elements.
Eg::) A = {1, 2, 3, 4} and B = {a, b, c, d} are equivalent sets.
Equal sets are always equivalent sets, but equivalent sets need not be equal sets.
If two sets A and B are said to be equivalent we can write it as A ≈ B
(4) SUBSET, PROPER SUBSET and SUPER SET -
Subset : - Two sets A and B such that every element of A is also an element of B, then we can say that A is a subset of B.
Superset : - When A is a subset of B, or the elements of A is contained in elements of B, then we can say that B is Superset of A.
Every set is a subset of itself.
Null set is a subset of every set.
(5) POWER SET -
The family of all subsets of set A is called the power set of A and is denoted power set of A and is denoted by P(A).
Eg::) If A = {1, 2, 3}, then subsets of set A are {1}, {2}, {1, 2}, ø
P(A) = { ø, {1}, {2}, {1, 2} }
n[p(A)] = 4 = 22
In general A is a set with n(A) is m , then it can shown that n[p(A)] = 2m > m = n(A).
(6) UNIVERSAL SET -
A superset which consisting of all other sets in the pattern. Such a superset is called Universal Set. And it is denoted by U.
Eg::) Universal set is a real numbers, R in which even integers, prime numbers, positive integers lie.
(1) Even natural numbers less than 20, i.e., 2, 4, 6, 8, 10, 12, 14, 16, 18.
(2) The districts in Kerala.
In each of the above collections we can say that given object is belongs to or does not belong to particular set. So we can say that district "Trivandrum" belongs to the collections of districts in Kerala. Just look at the below example, for identifying whether the given collection is well defined or not.
(1) The collection of best horror novels in English Language.
In the above case the collection may vary from person to person. So we can not say it is well defined.
The objects which constitute the given set, are called elements. The elements of a given set are also known as members or points of a set.
Capital Letters A, B, C............., Y, Z are used to denote sets and small letters a, b, c, ....., y, z are used to denote elements of the set. If it is an element of the set S , then we can write it in symbol as "a Є S" and read it as " a is an element of S " or " a is a member of a S " or " a belongs to S ".
For specifying a set we choose two methods.
(1) Tabular or Roster Form :: -
In this form the set is represented by listing all its elements, separating the elements by commas and enclosing them in braces.
Eg::) Z = { 0, 2, 4, 6, 8} is a set of even numbers less than 10.
(2) Rule or Set Builder Form::-
In this form members of the set are represented by stating their characteristic property.
Eg::) Z = {x|x is an even number ≤ 10 } is the set of even numbers less than or equal to 10. Here " | " or " : " is used to mean "such that".
Types of Set::-
(1) The EMPTY Set -
A set which has no element is called the empty set ( null set or void set ) denoted by ø . Here we give some examples of empty set.
Eg::) If A = { x: 8 < x < 9, x is a natural number between 8 and 9 }, then A is an empty set, because there is no natural numbers between 8 and 9 .
(2) FINITE and INFINITE Sets -
A set consisting of a definite number of elements is called a finite set.
Eg::) Let P = {2, 3, 4, 5, 6} . Then Z is a finite set of natural numbers between 1 and 7 containing 5 elements.
A set having an infinite number of elements is termed an infinite set.
Eg::) Let P = {0, 1, 2, 3, 4, ..........} . Then Z is a infinite set of whole numbers.
The following symbols are used to denote the infinite sets of real numbers .
(1) N : the set of natural numbers.
(2) Z : the set of integers.
(3) Q : the set of rational numbers.
(4) R : the set of real numbers.
(5) Z+ : the set of positive integers.
(6) Q+ : set of positive rational numbers.
(7) R+ : the set of positive real numbers.
(3) EQUAL and EQUIVALENT SETS -
Equal Sets : - Two sets A and B are said to be equal , if they consists of exactly the same elements, and we say that A = B, otherwise that sets are said to be unequal.
Eg::) A = { x|x is a letter in the word ARYA }
B = { x|x is a letter in the word RAYA } are equal sets.
Equivalent sets : - The sets which have the same number of elements.
Eg::) A = {1, 2, 3, 4} and B = {a, b, c, d} are equivalent sets.
Equal sets are always equivalent sets, but equivalent sets need not be equal sets.
If two sets A and B are said to be equivalent we can write it as A ≈ B
(4) SUBSET, PROPER SUBSET and SUPER SET -
Subset : - Two sets A and B such that every element of A is also an element of B, then we can say that A is a subset of B.
Superset : - When A is a subset of B, or the elements of A is contained in elements of B, then we can say that B is Superset of A.
Every set is a subset of itself.
Null set is a subset of every set.
(5) POWER SET -
The family of all subsets of set A is called the power set of A and is denoted power set of A and is denoted by P(A).
Eg::) If A = {1, 2, 3}, then subsets of set A are {1}, {2}, {1, 2}, ø
P(A) = { ø, {1}, {2}, {1, 2} }
n[p(A)] = 4 = 22
In general A is a set with n(A) is m , then it can shown that n[p(A)] = 2m > m = n(A).
(6) UNIVERSAL SET -
A superset which consisting of all other sets in the pattern. Such a superset is called Universal Set. And it is denoted by U.
Eg::) Universal set is a real numbers, R in which even integers, prime numbers, positive integers lie.
