Sets Math

A set can contain anything, be it a collection of numbers, days of the week, fruits, etc. A simple example is a set of positive Integers up to 5, which looks like this: {1, 2, 3, 4, 5}. But exactly how do we define and use sets? Let's take a look.

What are sets?

Sets in mathematics are an organised collection of objects called elements. They are noted mathematically by curly brackets {}.

Elements of sets can be represented using various notations, the roster, or the set builder, and we will return to this later.

Symbols used in sets

Specific symbols are used to describe given sets. Below are common ones and their meaning.

Elements of a set

The items contained in a set are called the elements of the set. They are denoted by curly brackets with commas separating each element. We can use specific Notation to denote that something is an element of a particular set. For example, if we had A = {1, 2, 3, 4}, we could write that 3 ∈ A, which means 3 is an element of A. However, as it is evident that 5 is not a member of A, that can be denoted as 5 ∉ A.

• Here are examples of commonly used sets.

  • N - Set of all Natural Numbers

  • Z - Set of all Integers

  • Q - Set of all rational numbers

  • R - Set of all Real Numbers

  • Z + - Set of all positive integers

Order of sets

To define a set, it must be a collection of unique elements. A significant property of sets is that the elements somehow should be related to each other or share a common property. For example, by defining a list of primary colours in a set, we mean that all the elements are primary colours.

Cardinality denotes the total Number of elements in a set. This means that if we have a set of Natural Numbers below 6, the cardinality of that set will be 5. Take our set to be A = {1, 2, 3, 4, 5} – there are five elements present in the set. That makes our cardinality 5. The cardinality of A is denoted by | A | or n (A), where n is the number of elements in the brackets, and A is any given set.

Representation of sets

There are various ways sets can be represented. The fundamental difference is in the way the elements are listed. They can be represented by either semantic, roster, or set builder form.

Semantic form of representing sets

This Notation is a statement form of describing the elements of a set. For example, we can list natural Prime Numbers below 20. Another example is the list of the months in a year. In semantic form, they can even be written as {set of odd natural numbers less than 10}.

Roster form of representing sets

The roster form is the most commonly used notation used for sets. Elements are denoted by curly brackets and separated by commas. With this type of notation, the elements of the set are usually mentioned. For example, a set of odd natural numbers below 10 will be A = {1, 3, 5, 7, 9}. Equating A to our set means that anywhere we find A, we are talking about our list of odd natural numbers.

In another instance, you have a set with infinite elements, which is usually expressed with a series of dots at the end of the last stated element. For example, a set of positive integers will be denoted by $Z^{+}$ = {1, 2, 3, 4, 5, ....} This means there are endless numbers following 5 in the order that has been already expressed.

Set builder form of representing sets

This mathematical notation is used to describe sets by demonstrating the properties that its members must satisfy. There is usually a statement that specifically describes the common feature of all the elements of a set.

• For example, a set of positive integers up to 5 can be denoted by the set builder as $\{x|x\le 5\}$.

• Another example could be $A=\{x|xisanevennumber,x\le 12\}$. This notation states that all the elements of set A are even numbers that are less than or equal to 12. By writing this in roster form, we will have A = {2, 4, 6, 8, 10, 12}, and its cardinality will be 6.

Types of sets

There are many different types of sets in mathematics. We will go over them in this section.

Empty set

Sets that do not contain any elements are called empty sets or null sets. They are denoted by either {} or ∅.

Singleton set

These types of sets have only one element contained in them. They are also called unit sets. For example, A = {4}

Finite sets

Finite sets are sets with a countable number of elements in them. For example, A = {a set of positive integers below 7} will be A = {1, 2, 3, 4, 5, 6} or {x | x is positive integer <7}.

Infinite set

These are sets that contain an infinite number of elements. An example of this set is Z = {set of all integers}. Another example is multiples of three. Which can be denoted by C = {3, 6,9, 12, 15, .....}. The series of dots after the last element listed is used to express its infinite status.

Equal sets

Two sets are said to be equal when they contain the same elements. The order in which they are arranged does not matter. For example, if I had two sets, A and B, where A = {2, 3, 4, 5} and B = {5, 4, 3, 2}, they are said to be equal.

Equivalent sets

When two sets contain the same number of elements even when the elements are different, they are considered equivalent. For example, A = {1,2,3,4} and B = {9, a, 3, w} are equivalent.

Disjoint sets

Two sets are considered disjoint if they do not contain a common element. For example, sets A and B are disjoint if A = {1, 2, 3, 4} and B = {7, 8, 9, 10}.

Subsets

Set A is considered a subset of B if all elements of A are present in set B. It is expressed mathematically by the notation A ⊆ B. By this definition, sets are considered subsets of themselves. For example, if B = {4, 6, 8,} and A = {6, 8}, A ⊆ B. When a set (A) is not a subset of another (B), it is denoted by A ⊈ B .

Empty sets are also regarded as subsets to every set. And empty sets have one subset, itself, whilst non-empty sets have at least 2 subsets, 0 and itself.

Proper subsets

If A ⊆ B, yet A ≠ B, then A is considered a proper subset of B. This can be denoted by A ⊂ B. For example, if A = {9, 12} and B = {3, 6, 9, 12}, then A ⊂ B.

Supersets

Set A is considered a superset of B if all elements of B are present in set A. it is denoted by the symbol ⊇. For example, if A = {1,2,3,4} and B = {1,2,3}, then A ⊇ B.

Universal sets

This is a set which contains elements of all related sets without repetition. It is denoted by the symbol U. For example, if A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then the universal set here is U = {1, 2, 3 , 4, 6, 8}.

Operating sets

Under certain conditions, operations of sets can be carried out in set theory. Some basic operations are:

• Union of sets

• Intersection of sets

• Complement of a set

• Cartesian product of sets

• Set difference

Union of sets

A union of sets contains all elements of the related sets. So if we have set A and B, a union will be all elements of A and B. It is denoted by the symbol U. Mathematically, a union of A and B will look like AUB.

Intersection of sets

An intersection set is one that contains common elements of related sets. An intersection of sets A and B will be elements that appear in both A and B. It is denoted by the symbol ∩. This means an intersection of sets A and B will mathematically be written as A ∩ B.

Complements of a set

Complement sets contain all elements in the universal set that are not in the given set. Assuming A is a subset of a much larger set called a universal set, the complement of A is all elements present in the universal set that aren't present in A. The complement will be denoted by A '.

The Cartesian Product of sets is defined as the set of all ordered pairs (x, y) from two sets, A and B such that x belongs to A and y belongs to B.

Set difference

Set difference lists the elements in set A that are not present in set B. It is denoted by A - B. For example, if A = {1, 2, 3, 4} and B = {1, 3, 5, 7 }, then A - B = {2, 4}.

Properties of sets

Sets, just like numbers, also have properties associated with them. The set formula is given in general as n (A∪B) = n (A) + n (B) - n (A⋂B), where A and B are two sets and n (A∪B) shows the number of elements present in either A or B and n (A⋂B) shows the number of elements present in both A and B. We will discuss six important properties given three sets A, B and C in this section.

Commutative property:

• AUB = BUA

• A ∩ B = B ∩ A

Associative property:

• (A ∩ B) ∩ C = A ∩ (B ∩ C)

• (AUB) UC = AU (BUC)

Distributive property:

• AU (B ∩ C) = (AUB) ∩ (AUC)

• A ∩ (BUC) = (A ∩ B) U (A ∩ C)

Identity property:

• AU ∅ = A

• A ∩ U = A

Complement property:

• AUA '= U

Idempotent property:

• A ∩ A = A

• AUA = A

Worked examples of sets

Here are a few worked examples on sets.