Set Notations

A set is a collection of well-defined objects that are referred to as elements. An element is an object that belongs to a set, and a set is a collection of elements. A set is a group of distinct elements of the same kind; for example, a basket of apples, a set of teacups, and a set of real numbers or natural numbers. 

Set builder method

The study of sets, as well as the relationships between them, is called set theory. Moving further, set notations can be taken as a system of symbols that helps to define elements of sets, defining relationships among different types of sets, algebraic properties of sets, and defining operations among sets. 

Set Builder Notation – Elements of a set

The infinite number of elements of a set can be represented by the set builder notation. Set builder notation can represent real numbers, integers, and natural numbers. Also, a set with an interval or equation can be best described by this method.

Sets are usually represented by capital letters. An element (or member) of a set belongs to the set, and the set contains the elements.

For example:

A = {1, 2, 3, 4, 5}

A is the set, and 1, 2, 3, 4, 5 are the elements or members of the set. Set elements can be written in any order but cannot be repeated. When using alphabets, the set elements are all represented in small letters.

Here are a few 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

Types of sets and relationships among sets

This section will cover different types of sets, how they are related, and how to represent them.

Set builder notation – Finite set

A finite set consists of a definite number of elements. If the elements of this set have an indefinite number of members, the process will run out of elements to list.

Example: A set of all English alphabets (because it is countable).

Set builder notation – Infinite set

An infinite set has an uncountable number of elements and cannot be represented as a roster. Thus, it is also known as an uncountable set.

Example: The set of all integers or a set of whole numbers.

Empty set in set builder form

The empty set has no elements. The cardinality (number of elements) of an empty set is 0, so it is a finite set. A set is infinite if the number of elements in it is infinite, whereas a finite set contains a finite number of elements.

The empty set in set builder form is denoted by the symbol “Φ” and “φ” or {}.

Set Builder Notation – Singleton set

Sets with exactly one element are called singletons or unit sets. For example, set [null] is a singleton because it contains the element null.

Example: Set X ={2} is a singleton set.

Operations of sets

Set operations combine elements of two or more sets, according to the operations performed on them. The three major types of operations performed on sets include:

• Union of sets
• Intersection of sets
• Difference of sets

Let us briefly learn about these operations and the set builder notation for each of the set’s operations.

• Union of sets

Assuming two sets A and B, the union of the two sets will contain all the elements of both sets.

Formula:

A ∪ B = {x: x ∈ A or x ∈ B}, where x is the element present in the sets A and B.

Example: If set A = {p, q, r, s} and set B = {x, y}

Then, the union, A ∪ B = {p, q, r, s, x, y}

• Intersection of sets

Given two sets, A and B, the intersection is the subset of the universal set U that contains elements common to both sets A and B. It is denoted by the symbol ‘∩ ‘.

Formula: A∩B = {x : x ∈ A and x ∈ B}

where x is the common element of both sets A and B.

n(A∩B) = n(A) + n(B) – n(A∪B)

Example: A = {a, b, c} and B = {b, d, f}

Then, A∩B = {b}; because b is common to both the sets.

• Difference of sets

A set that consists of elements present in A but not in B is represented by A-B. Consider two sets A and B; the difference between these two sets is equal to the set made up of elements present in A but not in B.

The difference between sets A and B equals the intersection of set A with the complement of set B.

Formula: A−B=A ∩ B’

Example: If A = {a, b, c, d, e, f, g} and B = {f, g}

then, the difference of set A and set B is

A – B = {a, b, c, d, e}

Algebraic properties of sets

A set algebra develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation, as well as the relations of set equality and set inclusion. These procedures are also used to evaluate expressions and perform calculations involving these operations and relations.

• Commutative property

The addition and multiplication of sets are commutative, and changing the order of the sets in the problem will not alter the answer. Subtraction and division are NOT commutative.

Let us consider there are two sets A and B:

Formula:

A∩B=B∩A   (intersection)

A U B = B U A (union)

Example: A = {a, b, c} and B = {b, d, f}

so A∩B={b} and B∩A={b}

A U B=B U A={a,b,c,d,f}

• Associative property

The associative property explains how addition and multiplication are possible irrespective of how the sets are grouped.

Any relation with three or more sets will have the same result regardless of how the elements are grouped.

Formula: (A ∪ B) ∪ C = A ∪(B ∪ C) 

Example: A={a,b} B={c,d} C={e}

so (A ∪ B) ∪ C = A ∪(B ∪ C)={a,b,c,d,e}

• Distributive property

The distributive property demonstrates that it is possible to multiply a set by multiplying each addend separately. The numbers in the brackets are distributed for each number outside the brackets.

It is one of the most frequently used properties in Mathematics.

Formula: A ∪ (B ∩ C) = (A∪B) ∩( A∪C) 

Example: A={a,b,c} B={b,e} C={e}

so A ∪ (B ∩ C) = (A∪B) ∩( A∪C) ={a,b,c,e}

•  Power set

A power set consists of all subsets, including the empty set and the original set itself. It is denoted by the letter P.

Formula:

The power set P(A)={{}, { a }, { b }, { c }, {a,b}, {b,c}, {c,a}, {a,b,c}}

Now, the power set has 23 = 8 elements.

Example: A = {1, 2} , the power set P(A) = {{}, {1}, {2}, {1, 2}}.

Properties of power set

• In A’s power set, there are 2n elements, where n is the number of elements in A
• A countable finite set has a power set that is countable and much larger than the original set.
• P(S) can be mapped one-to-one to the real numbers for a set of natural numbers.
• As an example of Boolean Algebra, P(S) of set S results from the union, intersection, and complement of sets.

Conclusion

With this discussion, you have a clear idea of sets, how to represent sets or set builder notations, different elements of sets, and related formulas. Sets are easy to learn, but sometimes, understanding their differences becomes a challenge. So, this article will give you a clear idea of different types of sets and how they differ from each other.