Introduction of Sets

 

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Introduction of Sets

Set Theory, branch of mathematics concerned with the abstract properties of sets, or collections of objects. A set can be a physical grouping, such as the set of all people present in a room; or a conceptual aggregate, such as the set of all British prime ministers, past and present. Each of these sets is defined by a property that its members share, but it is possible for a set to be a completely arbitrary collection.

Use of the word set as a formal mathematical term was introduced in 1879 by Georg Cantor (1845–1918). For most mathematical purposes we can think of a set intuitively, as Set theory was first given formal treatment by the German Mathematician Georg Cantor in the 19th century. Cantor did, simply as a collection of elements. 

For instance, if C is the set of all countries that are currently in the United Nations, then the United States is an element of C, and if I is the set of all integers from 1 to 100, then the number 57 is an element of I. The set concept is one of the most basic in mathematics, explicitly or implicitly, in every area of pure and applied mathematics, as well as Computer science. Relationships between elements of sets occur in many contexts. We deal with many relationships such as student’s name and roll no., teacher and their specialisation, a person and a relative (brother – sister, mother – child etc.) In this section, we will discuss mathematical approach. To the relation. These have wide applications in Computer science (e.g. relational algebra). 

Definition of Sets

A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. 

Examples of sets are:

• A set of rivers of India.

• A set of vowels.

A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set. 

capital letters, A, B, X, Y, . . . , to denote sets, and lowercase letters, a, b, x, y, . . ., to denote elements of sets. 

Membership in a set is denoted as follows:

a ∈ S denotes that a belongs to a set S

a, b ∈ S denotes that a and b belong to a set S

Representation of a Set

Sets can be represented in two ways −

• Roster or Tabular Form

• Set Builder Notation

Roster or Tabular Form

The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.

Example 1 − Set of vowels in English alphabet, A={a,e,i,o,u} A={a,e,i,o,u}

Example 2 − Set of odd numbers less than 10, B={1,3,5,7,9}B={1,3,5,7,9}

Set Builder Notation

The set is defined by specifying a property that elements of the set have in common. The set is described  as {x: x satisfies properties P}. and read as ‘the set of those entire x such that each x has properties P.’

Example 1- If B= {2, 4, 8, 16, 32}, then the set builder representation will be: B={x: x=2n, where n ∈ N and 1≤ n ≥5}

Example 2 − The set {a,e,i,o,u} is written as −

A={x:x is a vowel in English alphabet}

Example 3 − The set {1,3,5,7,9} is written as −

B={x:1≤x<10 and (x%2)≠0}

If an element x is a member of any set S, it is denoted by x∈S and if an element y is not a member of set S, it is denoted by y∉S.

Standard Notations

• x ∈ A x belongs to A or x is an element of set A.
• x ∉ A x does not belong to set A.
• ∅ Empty Set.
• U Universal Set.
• N The set of all natural numbers.
• I The set of all integers. 

Cardinality of a Sets

The total number of unique elements in the set is called the cardinality of the set. The cardinality of the countably infinite set is countably infinite. Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.

Example –|{1,4,3,5}|=4,|{1,2,3,4,5,…}|=∞

Examples :

• Let P = {k, l, m, n}

The cardinality of the set P is 4.

• Let A is the set of all non-negative even integers, i.e.

A = {0, 2, 4, 6, 8, 10……}.

As A is countably infinite set hence the cardinality.