What Are Sets

Sets are group of things that have some properties in common. This property or properties define the requirement for anything to be a member of that set or not. Sets are usually denoted by uppercase letters.

Let's take some examples of sets.

If we say that set $A$ is a set of polygons of $6$ or less sides. This means that all triangles, all quadrilaterals, all pentagons as well as all hexagons are members of this set. But polygons of $7$ sides or $8$ sides or more are not members of this set.

If we say that set P is a set of all four legged mammals, then the set will contain dogs, cats, goats and all other mammals that have four legs. However, human beings, monkeys or fish will not be members of this set.

Each member of a set is called an Element.

We usually write the set elements using the curly braces. If we define our set $P$ as the set of all individuals winning more than one Nobel prize (there are 4 such individuals), we can write

$P = \{Marie\ Curie,\ John\ Bardeen,\ Linus\ Pauling,\ Frederick\ Sanger\}$

or, if we define a set $A$ containing all even prime numbers, we can write

$A = \{2\}$

This set contains only one element.

Can a set be empty? Yes, it can. Let's see how.

We define a set $B$, of numbers that are divisible by $10$ but not divisible by $5$. We know that there is no such number, so we can say the set $B$ is empty. We denote this by writing:

$B = \{\}$

or, also as,

$B = \phi$

($\phi$ is the symbol for the Greek letter $\unicode{0x201C}phi\unicode{0x201D}$)

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We can also say that $B$ is a null set.

Order of elements in a set:

Order in which the elements are written in a set has no significance. It can be written in any order.

So, if there is a set $S$ containing Cat, Dog, and Mouse we can write

$S = \{Cat,\ Dog,\ Mouse\}$

or, we can also write

$S = \{Mouse,\ Dog,\ Cat\}$

or any other order that you may like. These are treated as the same set.

There are a few things we need to remember about sets.

- The property or properties that define the set must be clearly defined and distinguishable.

- Each element of any set should be a distinct element, which means, no element can be repeated within a set.

We will take some example of invalid sets to illustrate this.

Lets try to define a set $A$ of all tasty food in the world. This is not a valid set, because tasty is something that varies from person to person. A food which is tasty for somebody, may be disliked by others. Therefore, this is not a valid set definition.

If we say, set $Y$ is the set of all popular sports in the world. Popularity does not have a clear definition and cannot be measured in a definitive way. For example, soccer may be popular in some countries and not so popular in other countries. So this is also an invalid set.

Let's say we define a set $P$ of all prime number less than $10$.

And if we write:

$P = \{2, 3, 3, 5, 7\}$

This is an invalid set, since the element $3$ occurs more than once in the set. Remember, no element can be repeated in a set.

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The correct representation of this set should be $P = \{2, 3, 5, 7\}$

We will take a moment to summarize what we just learnt:

- Sets are collection of things (numbers, objects, etc) that are identifiable by one or more well defined properties.

- Member of a set is called element.

- Order of elements in a set is not important.

- Set can be of 0 size (null set) or of infinite size.

- Sets are usually named with uppercase letters.

- Set elements are written with the curly braces: $\{ \}$.

- Elements cannot repeat in a set.