In the previous episode of Discrete Mathematics, we discussed about definition of mathematics. We understood structures and how the definition of mathematics is based on structures.

Today, in another episode of Discrete Mathematics we will explore Set; a mathematical structure.

What is Set?

A set is a mathematical structure that is a well-defined collection of distinct objects.

Let us understand it with an example. I will mention some statements and categorize them into sets or not. Let's kick off.

$$A\:set\: of\: planets\: in\:our\:solar\:system.$$

$$A\:set\:of\:student\:who\:scored\:over\:60\:mark\:in\:Discrete\:Mathematics.$$

$$A\:set\:of\:goal\:scorer\:in\:a\:footbal\:match.$$

In the above example, all the statements are set because we can define them. Similarly, below are examples of non-set.

$$A\:set\:of\:good\:teachers.$$

$$A\:set\:of\:hot\:stars\:in\:universe.$$

In the above example. all of the statements are non-set because they are not well-defined.

Since we have already understood with an example what sets are, allow me now to mention some standard sets.

• A set of natural numbers (N)

• A set of whole numbers (W)

• A set of integers (Z)

• A set of positive integers (Z+)

• A set of negative integers (Z-)

• A set of non-positive integers (Z - Z+)

• A set of non-negative integers. (Z - Z-)

• A set of rational numbers. (Q)

We have been representing sets with a straight sentence in the English language but wait...Is it the case that, we can only represent the sets the way we have done so far? Well, the answer is a big "NO"! There are three different ways of representing a set. The illustration below shows different ways of representing sets

We have three different ways to represent our set. The most formal method of representation is the property method/set-builder method.

Set-Builder Representation of set

$$S=x|\: x\:is\:a\:natural\:number\:less\:than\:5$$

Roster Representation of set

$$S = {1,2,3,4}$$

Well before wrapping up let me explain some more concepts or terminologies that are essential to understand.

Equal and Equivalent Set

We say two sets are equal if both sets have the same elements in them. Similarly, we say two sets are equivalent if both sets have the same cardinality.