Definition of Sets

Introduction

When we shop in a fruit market or supermarket, we see fruits grouped by type, such as grapes, mangoes, and oranges. In mathematics, a collection like this can be called a set if its members are clearly defined.

Not every collection qualifies as a set. The objects in the collection must be identifiable without ambiguity.

Definition of a Set

Examples of collections that are sets:

• the set of male students
• the set of four-legged animals
• the set of fruits whose names begin with the letter S

Examples of collections that are not sets:

• the set of school subjects students like
• the set of delicious foods
• the set of clever students

From these examples, we can state:

A set is a collection of objects that is clearly defined.

The objects inside a set are called its members or elements.

Symbols and Notation

General rules for writing sets:

1. set names usually use capital letters such as $A$, $B$, or $C$
2. the members are written inside curly braces { }
3. members are separated by commas

Examples:

Common number sets include:

• natural numbers
• whole numbers
• integers
• even numbers
• odd numbers
• prime numbers
• rational numbers
• irrational numbers
• real numbers

Ways to Describe a Set

A set can be written in several ways.

1. Using words

Example:

2. Using set-builder notation

Examples:

3. Listing all members

Examples:

Members of a Set

Suppose:

Then 1, 2, 3, 4, and 5 are members of $A$, written as:

The symbol $\in$ means “is a member of”, while $\notin$ means “is not a member of”.

Since 6 is not in the set $A$, we write:

Number of Members

If $A$ is a set, the number of its members is written as $n(A)$. This is called the cardinality of $A$.

• a set with finitely many members is a finite set
• a set with infinitely many members is an infinite set

Examples:

1. If $Z$ is the set of integers between 0 and 6, then

   $$Z = \{1,2,3,4,5\}$$

   so:

   $$n(Z)=5$$

2. If $N$ is the set of natural numbers, then

   $$N = \{1,2,3,4,5,\ldots\}$$

   so:

   $$n(N)=\infty$$

The Empty Set

Definition: The empty set is a set with no members. Its symbol is $\varnothing$ or { }.

The empty set is a subset of every set.

Examples:

1. Let $H$ be the set of Indonesian presidents whose names begin with P. Since there are none:

   $$H = \varnothing$$

2. Let $M$ be the set of months with exactly 20 days. Since no such month exists:

   $$M = \varnothing$$

Subsets

Consider:

Every member of $B$ is also a member of $A$, so:

But $B$ is not a subset of $C$, so:

Likewise:

In general:

• $A$ is a subset of $B$ if every member of $A$ is also a member of $B$
• $A$ is not a subset of $B$ if at least one member of $A$ is not in $B$

Number of Subsets

Observe this pattern:

• if $A=\{1\}$, then it has 2 subsets
• if $B=\{1,2\}$, then it has 4 subsets
• if $C=\{1,2,3\}$, then it has 8 subsets

So we get the formula:

Examples:

1. $P=\{a,b,c,d,e\}$, so $n(P)=5$. Therefore the number of subsets is:

   $$2^5 = 32$$

2. $K=\{x \mid x \text{ is a natural number less than } 7\} = \{1,2,3,4,5,6\}$, so $n(K)=6$. Therefore the number of subsets is:

   $$2^6 = 64$$

Universal Set

If we are discussing cars, buses, trains, and airplanes, then the overall context is transportation. In set theory, this context is called the universal set.

The universal set is usually denoted by $S$ or $U$.

Example:

Determine a possible universal set for:

Possible universal sets include:

• the set of whole numbers
• the set of natural numbers
• the set of integers
• the set of even numbers

Evaluation

Interactive exercises for this section are available in the original source.

Closing

This article introduced sets, set notation, the empty set, subsets, and the universal set. These are core ideas that support later topics in set theory and logic.