Set Theory Symbols

Set Theory is a branch of mathematics in which we learn about sets and their properties.  A set is defined by a collection of objects. The objects of a particular set are its elements. Set theory is the study of such sets, and the relation that connects the sets to their elements. Set Theory was first created by Sir Georg Ferdinand Ludwig Philipp Cantor, who was a German mathematician. The history and creation of set theory are quite different from the history of other areas of mathematics. It was created to be able to talk about collections of objects that represent a particular set. Set theory has turned out to be an extremely important tool for defining some of the most complex and significant mathematical structures.

Some of the Basic Examples of Set are

• The collection of all green bottles.

• The collection of negative numbers.

• The collection of people born before 1995.

• The collection of the greatest football players

• The number of oceans  in the world

• Number of students in a class

• Collection English vowels

All of the above collections are sets. However, the collection of the greatest football players is not well-defined. Usually, we restrict our focus to just well-defined sets.

How To Define Sets?

There are three ways in which you define sets:

• Descriptive form

• Roster(listing) form

• Set builder form

For example: Define / represent a set of odd natural numbers that lie between the interval of 2 to 12 in the descriptive, roster, and set builder form.

Descriptive Form

Example: The set of all odd counting numbers between 2 and 12.

Roster Form

In roster form, all the elements/objects of a set are listed, separated by commas and enclosed between curly braces { }. 

Example: Consider a set containing the leap years between the year 1995 and 2015, then using the roster form it will be represented as

A ={1996,2000,2004,2008,2012}

The order of elements in a set does not matter in the roster form, i.e the order can be ascending or descending. 

Also, frequency  is ignored while representing the sets. E.g. If  X  represents a set that contains all the letters in the word RIVERS, the proper Roster form representation would be

X ={R,I,V,E,S }= {S,E,V,I,R}  

X ≠ {R,I,V,E,R,S}

Set Builder Form

In set-builder form, all the objects have the same property. This property does not apply to the elements that do not belong to the set. 

Example: If set P has all the elements which are even prime numbers, then

P={ a: a is an even prime number}

where ‘a’ is a symbolic representation that is used to describe the element.

‘:’ means ‘such that’

‘{}’ means ‘the set of all’

So, P = { a: a is an even prime number } is read as ‘the set of all a such that a is an even prime number’. The roster form for this set P would be P = 2. This set has only one element. Such sets are called unit sets.

Another Example:

A = {x: x is a set of two-digit perfect square numbers}

How?

A = {16, 25, 36, 49, 64, 81}

As we can see, in the above-given example, 16 is a square of 4, 25 is square of 5, 36 is square of 6, 49 is square of 7, 64 is square of 8 and 81 is a square of 9 

Even though 4, 9, 121, etc., are also perfect squares, they are not elements of the set A because it is limited to only two-digit perfect squares.

Important Points to be Remembered:

1. Use curly braces to represent sets.

2. Use commas to separate set elements from each other.

3. The variable in the set-builder notation doesn’t necessarily have to be x.

4. The ellipses (. . .) are used to indicate a continuation of a pattern/ series established before the ellipses i.e. {1, 2, 3, 4 . . . 100}.

5. The symbol | is called “such that”.

Set Membership

• An element or member of a set is also known as an object of the set.

• The symbol ‘∈’ means “is an element of”.

• The meaning of the symbol ∈ “is not an element of”.

• Capital letters are used to denote sets and lowercase letters are used to represent its objects i.e. A = {2, 4, 6, 8}. Thus, a means x is an element of A.

Is 2 ∈ {2, 4,6, 8}? The answer is yes.

Is 2 ∈ {1, 3, 5, 7, 11}? The answer is no.

Important Sets to Remember

• N denotes Natural or Counting numbers: {1, 2, 3 . .}

• W denotes Whole Numbers: {0, 1, 2, 3 . . .}

• I denote Integers: {. ., -5, -4, -3, -2, -1, 0, 1, 2, 3 . .}

• Q denotes Rational numbers: {p/q | p, q ∈ I, q 6= 0}.

• R denotes Real Numbers: {x | x is a number that can be written as a decimal}.

• Irrational numbers: {x | x is a real number and it cannot be written as a quotient of integers}

• Examples are: π, √ 3, and 3√ 9.

• ∅ represents Empty Set: {}, the set that contains nothing.

• U represents the Universal Set: the set of all objects currently under discussion.

Note:

Any rational number can be written either as a terminating decimal (e.g 0.5, 0.333, or 0.8578966) or as a repeating decimal (e.g 0.333 or 123.392545).

The decimal digits of an irrational number never terminate and never repeat.

The set {∅} is not empty but is a set that contains the empty set.

Questions to Practice :

1. Is ∅∈ {a, b, c}?

2.  Is ∅∈ {∅, {∅}}?

3.  Is ∅∈ {{∅}}?

4.  Is 1/3 ∈ {x | x = 1/p, p ∈ N}?

Cardinality

• The cardinality of a set means the number of unique elements within the set.

• The symbol of cardinality is n (A) or |A|, where A is a set.

•  If the cardinality of a set is a whole number, then it is called a finite set.

• If a set is so large that its cardinality is uncountable, then it is called an infinite set.

Note:

We don’t need to consider the order or how many times an object is included in a particular set. Thus, {3, 1, 2, 4}, {2, 3, 4, 1}, and {1, 2, 2, 3, 3, 3, 4, 4, 4, 4} all describe the same set.

Question:

If A = {5, 7, 9, 11, 13}

B = {5, 10, 15 . . . 100}

C = {3, 5, 7, 9, . . .}

D = {1, 3, 3, 1, 2}

P = {x | x is odd, and x < 12}

Find:

n (A) = ?

n (B) = ?

n(C) = ?

n (D) = ?

n (P) = ?

Set Equality

The sets P and Q are equal (written P = Q), if and only if: 

i. Every element of P is an element of Q.

ii. Every element of Q is an element of P. In other words, set P and Q are equal if and only if they contain exactly the same elements

{x, y, z} = {z, x, y} = {z, y, y, y, x, x,}

{3} = {x | x ∈ N and 1 < x < 5}?

{x | x ∈ N and x < 0} = {y | y ∈ Q and y is irrational}

Venn Diagrams & Subsets

Universe: It is the set consisting of all elements under consideration for a particular situation. It is called the universal set and is denoted by ‘ꓴ’. Venn diagrams are used to represent sets and their relationships with each other. Here, X is a universal set represented by a rectangle. Sets are represented by shaded regions, circles or other shapes within the rectangle.

Complement of a Set

If a set contains some specific objects, then the complement of that will not contain those elements or objects. 

For example: If P= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A= {1, 2, 3, 4, 5}, then A’= {6, 7, 8, 9, 10}.

Question

Let Q = {1, 2, 3 . . . 8}, P = {1, 2, 5, 6}, and R = {2, 4, 5, 7, 8}

What is: P’, the complement of P?

What is: R’, the complement of R?

What is: Q’, the complement of Q?

What is: ∅’, the complement of ∅?

Types of Sets

Sets are categorized into different types, based on elements or types of elements in them. These different types of sets in basic set theory are

• Finite Set: In this set, the number of elements is finite

• Infinite Set: In this set, the number of elements is infinite

• Empty Set: It has no elements

• Unit Set: It contains one only element

• Equal Set: Two sets having same elements are equal sets

• Equivalent Set: Two sets having the same number of elements are equivalent set

• Power Set: A set of every possible subset.

• Universal Set: Universal set is the type of set that contains all the sets under consideration.

• Subset: When all the elements of set X belong to set Y, then X is a subset of Y

Subset

Set X is a subset of set Y if every element of X is also an element of Y and in other words, Y contains all of the elements of X which is denoted X ⊆ Y. 

R, S, and T are the sets that are below in the Venn diagram below, which are subsets: 

If U = {2, 3, 4 . . . 8}, R = {2, 5, 6}, S = {2, 4, 5, 7, 8}, and T = {2, 6}. What element(s) are in the area where all the sets get overlapped? What object(s) are in the area outside all the sets?

Set Equality and Proper Subsets

Sets P and Q are equal only if: 

i. P ⊆ Q and

ii. Q ⊆ P

Proper Subset: P ⊂ Q if P ⊆ Q and P = Q

Questions

Whether or Not a Subset?

Is the set on the left a subset of the set on the right?

a. {a, b, c} {a, c, d, f}

b. {a, b, c} {c, a, b}

c. {a, b, c} {a, b, c}

d. {a} {a, b, c}

e. {a, c} {a, b, c, d}

f. {a, c} {a, b, d, e, f}

g. X X

h. ∅ {a, b, c}

i. ∅ ∅

Points to Remember:

• Any set is a subset of itself and also a subset of the universal set.

• The empty set is a subset of all the sets as well as itself.

Questions:

Is the set on the left equal to, a proper subset of, or not a subset of the set on the right?

{1, 2, 3} I

{a, b} {a}

{a} {a, b}

{a, b, c} {a, d, e, g}

{a, b, c} {a, a, c, b, c}

{∅} {a, b, c}

{∅} {}

Power Set

S (A) is the set of all possible subsets of the set A. For example, if A = {0, 2}, then S (A) = {∅, {0}, {2}, {0, 2}}.

Find the power sets of the following and their cardinality.

S (∅) =

S ({a}) =

S ({a, b}) =

S ({a, b, c}) =

Is there a pattern?

Alternate Method of Generating Power Sets:

Under this method, a tree diagram will be used to generate S (A). Every element in the set is either in a particular subset, or it’s not.

The number of subsets of a set with cardinality n can be calculated by using the formula: 2n.

The number of proper subsets can be calculated by using the formula: 2 n – 1.

Set Operations

Four important set operations that are used commonly are given below:

• Union of Sets

• Intersection of Sets

• Complement of Sets

• Difference of Sets

Intersection

The intersection of two sets, X ∩ Y, is the set of elements common to both the sets, that is, X and Y: X ∩ Y = {XX ∈ X and x ∈ Y}. 

Therefore, for an object to be in X ∩ Y, it must also be a member of both the sets, X and Y.

Disjoint sets

Two sets which have no common elements in them are called disjoint sets. Their intersection is empty: X ∩ Y = ∅

Set Union

The union of two sets, X ∪ Y, is the set of elements belonging to either of the sets:

 X ∪ Y = {x|x ∈ X or x ∈ Y}. Therefore, for an object to be in X ∪ Y, it must be a member of either X or Y. The total shaded area covered by X and Y falls under XUY.

Set Difference

The difference of two sets, X − Y, is the set of elements belonging to set X and not to set Y: X − Y = {x|x∈ A and x ∈ B}

Note: 

Generally, X − Y = Y – X.

Also, X ∈ Y means x ∈ Y’. Thus, X − Y = {x|x∈ X and x ∈ Y’} = X ∩ Y’

Questions:

Find the Unions:

{a, b, c}∪{b, f, g} =

{a, b, c}∪{a, b, c} =

Find the Set Difference:

{ 5, 2, 3, 1, 4} − {2, 4, 6} =

{ 1, 3, 5} {2, 6, 4} − {1, 2, 3, 4, 5} =

Given the sets:

U = {1, 2, 3, 4, 5, 6, 9}

P = {1, 2, 3, 4}

Q = {2, 4, 6}

R = {1, 3, 6, 9}

Find Each of These Sets:

P ∪ Q =

P ∩ Q =

P ∩ U =

P ∪ U=

P’ =

P' ∩ Q =

P’ ∪ Q =

P ∪ Q ∪ R =

P ∩ Q ∩ R =

P’ ∪ Q’ =

P’ ∩ Q’ =

P ∩ (Q∪R) =

(P’∪R) ∩ Q =

P − Q =

Q − P =

(P − Q) ∪ R’ =

In case if U = {q, r, s, t, u, v, w, x, y, z}, A = {r, s, t, i, v}, and B = {t, v, x}

Fill up the Venn diagram to represent U, A, and B

Shade the area in the Diagram for: A’ ∩ B’ ∩ C

Shade the area in the Diagram

a. (A ∩ B)’

b. A’ ∪ B’

Write Whether the Following Sets are Disjoint?

{c, b, a} and {d, e, f, g} …

{a, b, c} and {a, b, c} …

{a, b, c} and {a, b, z} …

{a, b, c} and {x, y, z} …

{a, b, c} and ∅ …

For any A, A and ∅ …

For any A, A and A’ …

De Morgan’s Laws

For any sets P and Q,

(P ∩ Q)’ = P’ ∪ Q’

(P ∪ Q)’ = P’ ∩ Q’

Common Symbols used in Set Theory

Various symbols are used for representing common sets. They are given in the table below:

Table 1: Symbols denoting common sets

Other Notations

Set Theory Formulas

• n( A ∪ B ) = n(A) +n(B) – n (A ∪ B) 

• n(A∪B)=n(A)+n(B)   {when A and B are disjoint sets}

• n(U)=n(A)+n(B)–n(A∩B)+n((A∪B)c)

• n(A∪B)=n(A−B)+n(B−A)+n(A∩B) 

• n(A−B)=n(A∩B)−n(B) 

• n(A−B)=n(A)−n(A∩B) 

• n(Ac)=n(U)−n(A)

• n(PUQUR)=n(P)+n(Q)+n(R)–n(P⋂Q)–n(Q⋂R)–n(R⋂P)+n(P⋂Q⋂R) 

Conclusion

This article focuses on Sets and their types. It's important in mathematics to designate numbers in sets for easier calculations. Students can go through this document for a thorough understanding of the concept as well as to prepare for their exams.