Sets and their representations

Sets are an organized collection of objects and can be represented in set-builder form or roster form. Usually, sets are represented in curly braces {}, for example, A = {1,2,3,4} is a set.  

Definition of Sets

Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter. The number of elements in the finite set is known as the cardinal number of a set.

Elements of a Set

Let us take an example:

$A=\{1,2,3,4,5\}$

Set is represented by the capital letter.  Here, $A$ is the set and $\{1,2,3,4,5\}$ are the elements of the set or members of the set. The elements that are written in the set can be in any order but cannot be repeated. All the set elements are represented in small letter in case of alphabets.  Also, we can write it as $1\in A,2\in A$ ( $1$ belongs to $A$ , $2$ belongs to $A$ ) etc. The cardinal number of the set is 5. Some commonly used sets are as follows:

• $N:$ 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

Order of Sets

The order of a set defines the number of elements a set is having. It describes the size of a set. The order of set is also known as the cardinality.

The size of set whether it is is a finite set or an infinite set, said to be set of finite order or infinite order, respectively.

Representation of Sets

There are two methods of representing a set:

• Roster or tabular form
• Set-builder form

Roster form

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

Examples:

1.The set of all even positive integers less than 7 is described in roster form as $\{2,4,6\}$.

2.The set of all natural numbers which divide 42 is $\{1,2,3,6,7,14,21,42\}$.

Note In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as

$\{1,2,3,6,7,14,21,42\}$.

3.The set of all vowels in the English alphabet is $\{a,e,i,o,u\}$.

4.The set of odd natural numbers is represented by $\{1,3,5,......\}$.The dots tell us that the list of odd 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. For example, 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 $\}$

Here we describe the element of the set by using a symbol $x$ (any other symbol like the letters $y,z$ etc. could be used) which is followed by a colon $:$ After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces. The above description of the set $V$ is read as "the set of all $x$ such that $x$ is a vowel of the English alphabet". In this description the braces stand for "the set of all", the colon stands for "such that".

For example, the set $A=\{x:x$ is a natural number and $3<x<10\}$is read as "the set of all $x$ such that $x$ is a natural number and $x$ lies between $3$ and $10$." Hence, the numbers $4,5,6,7,8,9$ are the elements of the set $A$.

If we denote the sets described in 2 , 3, 4 above in roster form by $A,B,C$ respectively, then $A,B,C$ can also be represented in set-builder form as follows:

$A=\{x:x$ is a natural number which divides 42$\}$

$B=\{y:y$ is a vowel in the English alphabet$\}$

$C=\{z:z$ is an odd natural number$\}$

Problems

1. Write the set $\{{\frac {1}{2}},{\frac {2}{3}},{\frac {3}{4}},{\frac {4}{5}},{\frac {5}{6}},{\frac {6}{7}}\}$

Solution: Here each member in the given set has the numerator one less than the denominator. Also, the numerator begin from 1 and do not exceed 6.

Hence, in the set-builder form

the given set is $\{x:x={\frac {n}{n+1}}$ , where n is natural number and $1\leq n\leq 6\}$