SETS

The concept of a set is one of the most fundamental in mathematics. One defines the term set as ” a well-defined collection of objects. The objects that make up a set (also known as the set’s *elements* or *members*) can be anything: numbers, people, letters of the alphabet, other sets, and so on.

We assume that the word ” set” is synonymous with the words “collection”, ” aggregate”, ” class” and is comprised of elements. The words “element”, ” object”, ” member, are synonymous.

If is an element of a set , then we write and, say belongs to or is in or is a member of . If does not belong to , then we write . It is assumed here that if is any set and is any element, then either or and the two possibilities are mutually exclusive.

The following are some illustrations of sets.

- The collection of vowels in English alphabets. This set contains five elements, namely .
- The collection of first five prime natural numbers is a set containing the elements .
- The collection of all states in the Indian Union is a set.
- The collection of past presidents of the Indian Union is a set.
- The collection of cricketers in the world who were out for 99 runs in a test match is a set.
- The collection of good cricket players of India is not a set, Since the term “good player” is vague and not well defined

In this chapter we will have frequent interaction with some sets, so we reserve some letters for these sets as listed below

- : for the set of natural numbers
- : for the set of integers
- : for the set of all positive integers
- : for the set of all rational numbers
- : for the set of all positive rational numbers
- : for the set of all real numbers
- : for the set of all positive real numbers
- : for the set of all complex numbers

DESCRIPTION OF A SET

A set is often described in the following two forms i.e (i) Roster form or Tabular form (ii) Set-builder form

ROSTER FORM: In this form a set is described by listing elements, separated by commas, within braces .

For example: The set of vowels of English Alphabet may be described as .

*Note: *

*The order in which the elements are written in a set makes no difference. Thus, and denote the same set.**Also, the repetition of an element has no effect. For example, is the same set as .*

SET BUILDER FORM: In this form, a set is described by a characterizing property of its elements . In such a case the set is described by or, , which is read as ‘the set of all such that holds’. The symbol or is read as ‘such that’.

The set of all even natural numbers can be written as or

For example:

- The set can be written as
- The set of all real numbers greater than and less than can be described as

TYPES OF SETS

EMPTY SET: A set is said to be empty or null or void set if it has no element and it is denoted by . In the roster method is denoted by .

The empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

It follows from this definition that a set is an empty set if the statement is not true for any .

For example:

A set consisting of at least one element is called a *non-empty* or *non-void* set.

*Note: *

*If and are any two empty sets then if and only if is satisfied because there are no elements in either or to which the condition may be applied. Thus, . Hence, there is only one empty set and we denote it by .*

SINGLETON SET: A set consisting of a single element is called a singleton set.

For example: The set is a singleton set. Also is a singleton set

FINITE SET: A set is called a finite set if it is either void set or its elements can be listed (counted, labelled) by natural numbers , and the process of listing terminates at a certain natural number (say).

In other words, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting.

CARDINAL NUMBER OF A FINITE SET: The number of distinct elements in a finite set is called its cardinal number. It is denoted as and read as ‘the number of elements of the set’. For example: (i) Set has elements.

INFINITE SET: An infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are: the set of all integers, , is a countably infinite set; and the set of all real numbers is an uncountably infinite set.

EQUIVALENT SETS: Two finite sets and are equivalent if their cardinal numbers are same. i.e. .

EQUAL SETS: Two sets and are said to be equal if every element of is a member of , and every element of is a member of .

If sets and are equal, we write and when and are not equal.

It follows from the above definition and the definition of equivalent sets that equal sets are equivalent but equivalent sets need not be equal. For example, and are equivalent sets but not equal sets.

SUBSETS

Let and be two sets. If every element of is an element of , then is called a subset of .

If is a subset of , we write which is read as is a subset of or ‘ is contained in, .

Thus, if and only if .

If is a subset of , we say that contains or, is a super set of and we write .

If is not a subset of , we write .

Obviously, every set is a subset of itself and the empty set is subset of every set.

*Note: Satisfies transitivity i.e. and *

A *proper subset* contains some but not all of the elements of the original set. is called a *proper subset* of If and then . In such a case, we also say that . Thus, if is a proper subset of , then there exists an element such that .

An *improper subset* is a subset containing every element of the original set.

The *empty set* is a proper subset of a given set.

It follows immediately from this definition and the definition of *equal sets* that two sets and are equal if and only if and .

Thus, whenever it is to be proved that two sets and are equal, we must prove that and .

SOME RESULTS ON SUBSETS

THEOREM 1: Every set is a subset of itself.

Proof: Let, be any set. Then each element of is clearly in itself. Hence, .

THEOREM 2: The empty set is a subset of every set.

Proof: Let be any set and be the empty set. In order to show that , we must show that every element of is an element of also. But contains no elements. Hence, every element of is in . Hence .

THEOREM 3: The total number of subsets of a finite set containing elements is .

Proof: Let be a finite set containing elements. Let . Consider those subsets of that have elements each. We know that the number of ways in which elements can be chosen out of elements is . Therefore, the number of subsets of having elements each is . Hence the total number of subsets of is

SUBSETS OF THE SET OF REAL NUMBERS

Following sets are important subsets of the set It of all real numbers:

- The set of all natural numbers
- The set of all integers
- The set of all rational numbers
- The set of all irrational numbers. It is denoted by Thus,

Clearly . and

INTERVALS AS SUBSETS OF

On real line various types of infinite subsets are designated as intervals as defined below:

CLOSED INTERVAL Let and be two given real numbers such that . Then, the set of all real numbers such that is called a closed interval and is denoted by .

Thus

OPEN INTERVAL lf and are two real numbers such that , then the set of all real numbers satisfying is called an open interval and is denoted by or .

Thus

SEMI-OPEN or SEMI-CLOSED INTERVAL lf and are two real numbers such that , then the sets and are known as semi-open and semi closed intervals. and are also denoted by and respectively.

The number is called the length of any of the intervals and .

These notations provide an alternative way of designating the subsets of the set of all real numbers. For example, the interval denotes the set of all non-negative real numbers, while the interval denotes the set of all negative real numbers. The interval denotes the set of all real numbers.

UNIVERSAL SET

A super set of each of the given sets. Such a set is called the *universal set* and is denoted by . Thus, a set that contains all sets in a given context is called the universal set.

If and , then can be taken as the universal set.

POWER SET

Let be a set. Then the collection or family of all subsets of is called the *power set* of and is denoted by .

That is

Since the empty set and the set itself are subsets of and are therefore elements of . Thus, the power set of a given set is always non-empty.

Let . Then, the subsets of are and . Hence,

We know that a set having elements has subsets. Therefore, if is a finite set having elements, then has elements. The cardinal number of is .

VENN DIAGRAMS

In Venn-diagrams the universal set is represented by points within a rectangle and its subsets are represented by points in closed curves (usually circles) within the rectangle.

- If a set is a subset of a set , then the circle representing is drawn inside the circle representing as shown in Figure.
- If and are not equal but they have some common elements, then to represent and we draw two intersecting circles as shown in the figure.
- Two disjoint sets are represented by two non-intersecting circles as shown in the figure.

OPERATIONS ON SETS

UNION OF SETS: Let and be two sets. The union of and is the set of all those element, which belong either to or to or to both and .

We shall use the notation (read as ) to denote the union of and .

Thus or

Clearly or or . And and

In the figure below, the shaded part represents . It is evident from the definition that .

If and are two sets such that , then . Also, , if .

lf and , then

If is a finite family of sets, then their union is denoted by or

INTERSECTION OF SETS Let and be two sets. The intersection of and is the set of all those elements that belong to both and .

Intersection of and is denoted by (read as ).

Thus and

Clearly, and

In the figure shown, the shaded region is . Evidently, and

If and are two sets, then if and if .

If is a finite family of sets, then their intersection is denoted by or

If , and , then . Therefore

DISJOINT SETS Two sets and are said to be disjoint, if . If then and are said to be’intersecting or overlapping sets.

DIFFERENCE OF SETS Let and be two sets. The difference of and , written as , is the set of all those elements of which do not belong to .

Thus and or .

Clearly, and

Similarly,the difference is the set of all those elements of that do not belong to i.e

As shown in the figure, the shaded part represents and also .

SYMMETRIC DIFFERENCE OF TWO SETS If and are two sets, the symmetric difference of set and set is the set and is denoted by .

COMPLEMENT OF A SET Let be the universal set and let be a set such that . Then, the complement of with respect to is denoted by or or and is defined the set of all those elements of which are not in .

Thus . Clearly,

Let the set of natural numbers be the universal set and let . Then .

LAWS OF ALGEBRA OF SETS

THEOREM 1 (Idempotent Laws) For any set (i) and (ii)

THEOREM 2 ( Identity Laws) For any set , (i) and (ii) . i.e. and are identify elements for union and intersection respectively.

THEOREM 3 (Commutative Laws) For any two sets and (i) and (ii) i.e. union and intersection are commutative.

THEOREM 4 (Associative Laws) If and are three sets, then (i) and (ii)

THEOREM 5 (Distributive Laws) If and are three sets, then (i) and (ii) i.e. union and intersection are distribution over intersection and union respectively.

THEOREM 6 (De-Morgan’s Laws) If and are any two sets, then (i) and (ii)

MORE RESULTS ON OPERATIONS ON SETS

THEOREM 1 If and are any two sets, then

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

THEOREM 2 If and are any three sets, then prove that:

(i)

(ii)

(iii)

(iv)

SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS

If and are finite sets, and be the finite universal set, then

(i) or,

(ii) are disjoint non-void sets.

(iii) i.e.

(iv) No. of elements which belong to exactly one of or

(v)

(vi) Number of elements in exactly two of the sets

(vii) Number of elements in exactly one of the sets

(viii)

(ix)