The foundation of set theory was laid by the eminent German Mathematician **Georg Cantor **during the latter part of the 19th century.

The following are some of the problems we shall pursue in this chapter:

- Find the number of positive integers ≤ N and divisible by a, b, or c.
- How many subsets does a finite set with n elements have?
- How would you define the set of legally paired parentheses?
- How many sequences of legally paired parentheses can be formed using n pairs of left and right parentheses?

**Set** is a collection of well-defined objects, called **elements**(or **members**) of the set.

There should be no ambiguity in determining whether or not a given object belongs to the set. It should be clear and non-debatable.

For example, the vowels of the English alphabet form a (well defined) set, whereas beautiful cities in the United States do not form a set since its membership will be debatable.

Sets are denoted by CAPITAL LETTERS and their elements by lowercase letters.

x\,\in\,A

Object x is an element of set A.

## Two Methods of Defining Sets:

## 1. Listing Method

A set can sometimes be described by listing members within braces. Example:

A\,=\,\{apple, grapes,banana,melon\}

Set A of fruits.

The order of the elements doesn’t matter, and repeated elements is only counted as one.

For example:

\{x,x,y,x,y,z\}\,=\,\{x,y,z\}

A set with a large number of elements that follow a definite pattern is

often described using ellipses `(…)`

by listing a few elements at the beginning.

For example, the set of letters of the alphabet can be written as `{a, b, c , . . . , z}`

and the set of odd positive integers as `{ 1, 3, 5,… }`

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