Set Notations

Semantic Form

The simplest way to represent a set is a statement to describe the elements it is comprised of. These can be simple phrases.

Example

• collection of all wooden chairs
• collection of real numbers between 0 and 1

Roster Form

One way of describing a set that has a finite number of elements is by listing them between braces $\{\}$.

Example

• set of all positive integers less than four
  • $A=\{1,2,3\}$
• set of all vowels in the English alphabet
  • $V=\{a,e,i,o,u\}$

When the general pattern of the elements is obvious, all members of the set may need not be explicitly described; instead, ellipses ($\dots$) can be used.

Example

• set of positive integers less than 100
  • $P=\{1,2,3,\dots ,99\}$

Set Builder Form

If a set is a large finite set or an infinite set , we describe it by specifying a property that the elements of the set have in common using the form $\{x|P(x)\}$. The vertical bar | used in this notation stands for “such that”. Sometimes a colon : is used in the place of the |.

Example

• $B=\{x\mid x\text{ is a positive integer less than 4 }\}$
• $C=\{x\mid x\text{ is a Computer Science student }\}$