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,… }.
