Set Theory

Introduction to Set Theory

In Math, sets are used to define the concept of functions and vector spaces. The original set theory was first introduced by mathematician Georg Cantor.

In math sets is a well-defined collection of objects.

Examples of sets can be: Even natural numbers less than 10, states of USA, all vowels in the word “Mississippi” etc.

The word well-defined implies that we have a defined rule which helps us to decide whether or not a particular object belongs to a given collection. So in the above examples we know that 7 does not belong to the set of even integers less than 10 but 2 would belong to that collection; or that Paris would not be included in the set containing states of the USA and the vowel ‘a’ would not be in the set of vowels of word ‘Mississippi’ in fact that collection would contain only vowel ‘i’.

However, if we define a collection such as ten most beautiful cities of the world then it would not be called as a set as it is not well defined, such a criterion of determining a city as beautiful may vary from person to person. Thus it is not a well-defined collection.

Set Theory Notation and Set Theory Symbols

Math Sets are usually denoted by a capital letter such as A, B, C etc. while the elements of a set are usually represented by lower case letters such as a, b, c etc.

If ‘a’ is an element of set A then we say that ‘a belongs to A’. The Greek symbol ∈ (epsilon) is used to denote the phrase “belongs to”. Thus we can write a∈A .

If ‘b’ is not an element of A we write b ∈B and read “b does not belong to A”.

If E is the set of even natural numbers less than 10 then, 2 ∈E but 7 ∈E

There are two methods of representing a set:

Set Roster Notation (or tabular form sets):

In set roster notation, all elements of a set are listed, the elements being separated by a comma and enclosed within braces { }.

Examples of set roster notation can be:

The set E of all even natural numbers less than 10 is described in the roster set form as E = {2, 4, 6, 8}

The set V of all consonants in the word MISSISSIPPI is described in the roster set notation form as: V = { M, S, P }

If the set contains an infinite number of elements then it is represented in the form of three dots as:

e.g. The set of all multiples of 5 is represented as {5, 10, 15, 20, 25, …} the three dots imply that the list continues indefinitely.

In the set roster notation, the order in which elements are listed is immaterial. Thus E can also be listed as {8,4,6,2} or {4,8,2,6}.

While writing the elements in a roaster set form, all elements are taken as distinct and no element is repeated. For example, the set of letters forming the word ‘MISSISSIPPI’ is {M,I,S,P} or {P,M,I,S} or { I, S,M,P}.

Set Builder Notation :

A set builder notation is defined as that form of writing a set in which all elements of a set possess a single common property which is not possessed by an element outside the set.

For example, in the set { 2, 4, 6, 8}, all elements possess a common property, namely each of them are even natural numbers less than 10 and no other number possess this property. Thus we can write this set in set builder notation as : E = { x : x is an even natural number and x < 10}

Similarly the set builder notation, A = { a : a is an alphabet in the word MISSISSIPPI} is read as “the set of all a such that a is an alphabet in the word MISSISSIPPI”. The braces stand for ‘set of all’ the colon stands for ‘such that’.

Null Set

Null Set definition: A set which does not contain any element is called the null set or the empty set or void set.

Null Set Example: P = { p : 5 < p < 6, p is an integer}

P is a null set or an empty set, because there is no integer between 5 and 6.

The null set is denoted by the symbol {} or φ.

Finite Sets: A set which is empty or consists of a definite number of elements is called a finite set, otherwise the set is called an infinite set.

By the number of elements in a set S, we mean the number of distinct elements of the set. We denote the number of elements in a set S by n(S). This number n(S) is also called the cardinal number of a set.

If n(S) is a natural number, then S is a non-empty finite set or definite set.

Example of cardinal number of a set is :

if A = { a : a is an alphabet in the word MISSISSIPPI} then n(A) = 4 or

if P = { p : 5 < p < 6, p is an integer} then n(P) = 0 .

However, the set X = { x : x is a multiple of 5} has an infinite number of elements and is thus an infinite set.

When an infinite set is to be represented in a roster set notation, we write a few elements of the set to indicate the defining property of the set followed (or preceded) by three dots.

Example of roster set notation of infinite set:

N = { n : n is a natural number } = { 1,2,3,4, … }

W = { w : w is a whole number} = { 0, 1, 2, 3, …}

Z = { z : z is an integer} = { …, -4 , -3, -2, -1, 0, 1, 2, 3, 4, …}

It is not always possible to denote every infinite set in a roster set notation.

For example, the set of all rational numbers Q cannot be described in a roster set notation because the elements of this set don’t follow any particular pattern as Q= {p/q ∶p,q ∈Z,q ≠0}

Similarly the set of all irrational numbers cannot be described in a roster set notation as Q’: the set of all irrational numbers = {x∶x∈R and x ∈ Q} where R denotes the set of real numbers.

Equal Sets

Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise the sets are said to be unequal and we write A ≠ B.

Example of equal sets can be: Let A = {4,6,8,10,12} and B = { 8,4,10,6,12} . Then A = B

Let A = {x: x is a multiple of 7 and x < 35} and B = {7, 14, 21, 28} then A = B

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Important topics in Set Theory