Sets And Their Representations

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Introduction to sets and their representations

Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.

The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”.

What is a Set?

A Set is a collection of well defined objects.

Examples of sets :

(1) Various kinds of triangles.

(2) Positive even numbers less than 9, i.e; 2,4,6,8.

(3) The solution of the equation: x2 – 5x + 6 = 0, viz, 2 and 3.

(4) The vowels in the English alphabet, namely, a, e, i, o, u.

We note that each of the above example is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belongs to a given collection or not.

Examples of Sets Representations Used Particularly in Mathematics

N : Set of all Natural numbers
Z : Set of all Integers
Q : Set of all Rational numbers
R : Set of all Real numbers
Z+: ...
... Set of all positive Integers
Q+: Set of all positive Rational numbers
R+: Set of all positive Real numbers

Points to remember:

(1) Objects, elements and members of a set are synonymous terms.

(2) Sets are usually denoted by capital letters A, B, C, P, Q, R, X, Y, Z, etc.

(3) The elements of a set are represented by small letters a, b, c, p, q, r, x, y, z, etc.

If 'a' is an element of a set A, we say that “' a' belongs to A” the Greek symbol `in` (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a `in` A. If ‘b’ is not an element of a set A, we write b `!in`A and read “b does not belong to A”.

Sets and their Representations

Sets can be represented in two methods:

(i) Roster or Tabular form

(ii) Set-builder form.

ROSTER/TABULAR FORM:

In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

Example:

(1) The set of all vowels in the English alphabet is {a, e, i, o, u}.

(2) The set of even natural numbers is represented by {2, 4, 6, . . .}. The dots tell us that the list of even numbers continue indefinitely.

SET-BUILDER FORM:

In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

Example:

(1) In the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by V, we write

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

(2) The set of even natural numbers is represented by

A = {y : y is a even natural number}

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