Ways of Describing Sets

A set maybe described by:

1. Listing the components or elements of the set.

For example,

$P=\{2,3,5,7,11\}$$P=\{2,3,5,7,11\}$P={2,3,5,7,11}quadP=\{2,3,5,7,11\} \quad$P=\{2,3,5,7,11\}$ or $S=\{2,5,7,9,11\dots \}$$S=\{2,5,7,9,11\dots \}$S={2,5,7,9,11 dots}S=\{2,5,7,9,11 \ldots\}$S=\{2,5,7,9,11\dots \}$

All of S's absent members are symbolized by the three dots that follow the number five. This suggests there are other members of set $S$$S$SS$S$ for which we do not have a record. We can always identify a pattern that will lead us to the missing members.

For example, the missing members of the set $S=\{2,5,7,9\dots 19\}$$S=\{2,5,7,9\dots 19\}$S={2,5,7,9dots19}S=\{2,5,7,9 \ldots 19\}$S=\{2,5,7,9\dots 19\}$ are $11,13,15$$11,13,15$11,13,1511,13,15$11,13,15$, and 17 .

2. Providing a linguistic description of the set's constituents or a definition of a characteristic shared by its elements.

For example, the sets $\mathrm{P}$$\mathrm{P}$P\mathrm{P}$\mathrm{P}$ and $\mathrm{S}$$\mathrm{S}$S\mathrm{S}$\mathrm{S}$ above can be written as

$P=\{$$P=\{$P={P=\{$P=\{$ prime numbers less than 12$\}$$\}$}\}$\}$ and

$S=\{$$S=\{$S={S=\{$S=\{$ positive integers greater than 1$\}$$\}$}\}$\}$

3. By means of a set-construction notation.

For instance,

The set $P=\{2,3,5,7,11\}$$P=\{2,3,5,7,11\}$P={2,3,5,7,11}P=\{2,3,5,7,11\}$P=\{2,3,5,7,11\}$ can be represented by the symbols

$P=\{x:1<x<12$$P=\{x:1<x<12$P={x:1 < x < 12P=\{x: 1<x<12$P=\{x:1<x<12$, where $\mathrm{x}$$\mathrm{x}$x\mathrm{x}$\mathrm{x}$ is a prime number $\}$$\}$}\}$\}$

Furthermore, the set $M=\{1,2,3,4\dots 10\}$$M=\{1,2,3,4\dots 10\}$M={1,2,3,4dots10}M=\{1,2,3,4 \ldots 10\}$M=\{1,2,3,4\dots 10\}$ can be represented in the set-construction notation as:

$$M=\{x:1\le x\le 10,\text{where}x\text{is an integer}\}$$$$M=\{x:1\le x\le 10,\text{where}x\text{is an integer}\}$$M={x:1 <= x <= 10," where "x" is an integer "}M=\{x: 1 \leq x \leq 10, \text { where } x \text { is an integer }\}$$M=\{x:1\le x\le 10,\text{where}x\text{is an integer}\}$$

Likewise,

$$\mathrm{X}=\{a,e,i,o,u\}\text{can be represented in the set}-$$$$\mathrm{X}=\{a,e,i,o,u\}\text{can be represented in the set}-$$X={a,e,i,o,u}" can be represented in the set "-\mathrm{X}=\{a, e, i, o, u\} \text { can be represented in the set }-$$\mathrm{X}=\{a,e,i,o,u\}\text{can be represented in the set}-$$

construction notation as:

$X=\{x:x$$X=\{x:x$X={x:xX=\{x: x$X=\{x:x$ is a vowel in English alphabets $\}.$$\}.$}.\} .$\}.$

$x$$x$xx$x$ is used to represent any element of the set under investigation. You can also use other symbols like $\mathrm{k}$$\mathrm{k}$k\mathrm{k}$\mathrm{k}$ or $\mathrm{t}$$\mathrm{t}$t\mathrm{t}$\mathrm{t}$ if you choose.

For detailed maths tutorial on the above subject, watch the video below.