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A set is a collection of definite well defined objects of perception or thought. A set may also be thought of as grouping together of single objects into whole. Hence a set may be a bunch of grapes, a tea set or it may consist of geometrical points, or straight lines.

A set theory ought to be one of the major ideas in arithmetic. All the operations in set theory could be based on sets. Set ought to be a group of individual terms in domain. The universal set has the each & every element of domain.

Sets are of various types. Based on the type and number of elements contained in a set we can have different types of set.

For example, a set with no element is an empty set, where empty set is a type of set.

The various types of sets are discussed below.

Any set which is superset of all the sets under consideration is said to be Universal set. It is denoted by U or S.

For example, if we have 3 sets under consideration, say A = { 1 , 2 , 3 } , B = { 3 , 4 , 5 } and C = { 1 , 5 }.

Then the universal set for these sets will be S = { 1 , 2 , 3 , 4 , 5 }.

Union of any set with universal set is always universal set. i.e., A U S = S

For example, A = { 10, 20, 30 }, B = { 10 , 50 , 60 } and universal set S = { 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 }

A U S = { 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 } = U

Another point to be notes is that intersection of any set with universal set is always the same set i.e., A ? S = A.

Considering the same example as above we have

A ? S = { 10 , 20 , 30 }. = A.

A set which consists of only one element or member is known as singleton set. In other words a singleton set is a set with exactly one member. For example, A = { a }, here set A has only a single element ' a ' hence set A is a singleton set.

Another example of singleton set is X = { x : x is a positive integer and x < 2 }. Here set X is in set builder form. When it is converted into roster form it gives X = { 1 } because we have positive integers as 1,2,3,4,5…… and among these the integer less than 2 is 1. So set X contains exactly one element that is 1. Hence set X is a singleton set.

Now suppose we have a set P = { { 1 , 2 } } , note that this is a single set as it contains a single member which is a set. However that member set is not singleton.

A set consisting of a natural number of objects, i.e., in which number of elements is finite (countable), is said to be a finite set. In other words the set whose cardinality is finite is known as finite set. For example, consider set A = { 1 , 2 , 3 }, the cardinality of the set is 3 as it consists of 3 members. Since the numbers of elements of set are finite so set A is a finite set.

Another example of finite set is X = { x : x is a natural number less than 10 }.The given set is in set builder notion , when it gets converted into roster form it gives X = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }.Cardinality of set X is 10 which is countable. Hence set X is a finite set.

Practical example of finite set is set of vowels { a , e , i , o , u } it consists of 5 elements and hence it is a finite set.

If the number of elements in a set is infinite, the set is said to be an infinite set. In other words the set which consists of infinite (uncountable) number of elements is known as infinite set.

For example, set of all natural number is infinite set.

Similarly set of all rational number between 0 and 1 is given by,

A = { x: x € Q, 0<x<1 } an infinite set. Here Q represents set of all rational numbers.

Two sets A and B consisting of same elements are said to be equal sets. In other words if each element of set A is an element of set B and each element of set B is an element of set A , the sets A and B are said to be equal. It is denoted as A = B

For example, A = { 1 , 2 , 3 } and B = { 1 , 2 , 3 } so A = B because set A and B contains exactly same elements.

Suppose we have A = { 1 , 2 , 3 } and B = { 2 , 1 , 3 } then also A = B because members of both sets are exactly the same however their order is not same, but it doesn’t make any difference.

Two sets may be equal however they are in different notion. For example, consider X = { 1 , 3 , 5 } and Y ={ y : y is a odd natural number and y ? 5 }. Both these sets contains exactly same elements so X = Y.

If a set consists of no element, it is said to be an empty set. It is a unique set with no member. Empty sets are also known as null set or void set. It is denoted by ?. The common way to define a null set is,

?= { x: x ? x }, it means the set fi contains the element x such that x is not equal to x, which in turn is not possible, hence set does not contain any member. So it is a null set.

For example, A = { }

Another example of empty set is X = { x : x is a integer and 5<x<6 }

Integers are ....-3,-2,-1,1,2,3..... .Since there is no integer lying between 5 and the set X is a empty set.

Suppose A is a given set. Any set B, each of whose elements is also an element of A , is called contained in A and is said to be a subset of A.

The symbol ? stands for “is contained in” or “is subset of ”. Thus if “B is contained in A” or “ B is subset of A” , we write B ? A.

When B is a subset of A , we also say ‘A contains B’ or ‘A is the superset of B’. The symbol $\supseteq$ is read for “contains” so A $\supseteq$ B means “A contains B”.

For example, we have A = { 1 , 2 , 3 , 4 , 5 } and B = { 2 , 4 }, since all the elements of B are also in A so B is a subset of A.

If B is a subset of A and B is not equal to A, then B is said to be proper subset of A. In other words, if each element of B is an element of A and there is at least one element of A which is not an element of B, then B is said to be proper subset of A. “Is proper subset of ” is symbolically represented a by c.

The set of all subsets of a given set A, is said to be the power set of A. The power set of A is denoted by P(A). If a set has n elements then it has 2^{n} subsets and hence the cardinality of its power set is 2^{n}

For example, if the set A = { a , b , c } then its subset are ? , { a } , { b } , { c } , { a , b }, { a , c } , { b , c } , { a , b , c }.

Then P(A) = {? , { a } , { b } , { c } , { a , b }, { a , c } , { b , c } , { a , b , c }}

1). Find power set of –

X = { ? , 1 , {b}}

Solution: Since, the given set has 3 elements so its power set will have 2^{3} i.e., 8 elements.

Subsets of X are

? , {?} , {1} , {{b}} , { ? , 1 }, { ? , { b }} , { 1 , { b } } , { ? , 1 , { b } }

P(X) = { ? , {?} , {1} , {{b}} , { ? , 1 }, { ? , b }} , { 1 , { b } } , { ? , 1 , { b } }}

2) If A and B are two sets such that A** **? B and B ? A, prove that A=B

Solution: Suppose A ? B and x € A

Therefore, x € A hence x € B

But B ? A

Therefore x € A ox € B

It implies that, A= B

3) Categorize the following into null and singleton sets –

(i) A = { x : x € N, 3x^{2} - 14x - 5 =0}

(ii) B = {x : x^{3} =8, 2x = 5}

(iii) C = { x : x € N, x + 5 = 5 }

Solution: (i) Solving the equation 3x^{2} – 14x -5 = 0 we get x = 5, -1/3. But x € N . Therefore Only possible x belonging to A is 5.

Hence A = { 5 } which is singleton set.

(ii) Equation 2x = 5 gives x= 5/2 . But x = 5/2 does not satisfy the equation x^{3} = 8. Thus there is no x which satisfies both the equations. Hence there is no element in B, i.e., B is a null set.

(iii) Equation x+5 = 5 gives x= 0, but 0 ¢ N. Thus there is no element x belonging to C. Hence C is an empty set.