Set Theory

In the set theory, we will learn the following set concepts:

  • Set Definition
  • Set Notation
  • Types of Sets and
  • Operations on Sets

SET THEORY CONCEPT NO. 1:

SET DEFINITION. A set is a collection of well defined objects.

Examples: {1, 3, 5, 7, 9} is a set of positive odd numbers less than 10. The property that the numbers 1, 3, 5, 7, and 9 are odd is their being well defined. Africa, Asia, America, Australia, Europe are sets of continents in this world. All students in a class whose names begin with a consonant. The continent is a well-defined property shared by the lands of the above names.

Note:

  1. A set is denoted by braces {} within which the members of the set are written.
  2. The capital letters are used for denoting sets and small letters for members of the sets.
  3. Members/Elements of a set

The members of a set are called its elements. In the set {1, 3, 5, 7, 9}, the numbers 1, 3, 5, 7, 9 are called members of the set or more precisely elements of the set {1, 3, 5, 7, 9}. The symbol ‘∈’ denotes ‘belongs to’ or it stands for ‘element of’. Therefore we write 1 ∈{1, 3, 5, 7, 9}, which is read as ‘1 is an element of (or belongs to) the set {1, 3, 5, 7, 9}. Likewise, other elements of the set can be similarly denoted. The symbol ‘’ denotes “does not belong to” or ‘is not an element of’. So, we can translate 2 {1, 3, 5, 7, 9} as 2 is not an element of the set {1, 3, 5, 7, 9}, or 2 does not belong to the set {1, 3, 5, 7, 9}.

Note:

Let A = {1, 3, 5, 7, 9} and B = {5, 7, 9, 1, 3}.

Now are the two sets A and B different? No. Why? Because a set is the same irrespective of the order in which the elements are written.

SET THEORY CONCEPT NO. 2:

The denotation of a Set/Representation of a set. There are three ways of denoting a set.

Descriptive method. In this method, the set is described in complete words in a sentence.

Example 1:

The set A = {1, 2, 3, 4, 5… 100} can be described as “A is a set of all the consecutive integers from 1 to 100”.

Note:

The three periods after 5 up to 100 stands for all the integers between the two numbers.

Roster method

In this method, the set is represented by writing its elements inside braces { }

Example 1:

A = {3, 6, 9, 12, 15}, A is a set of the first five positive multiples of 3.

X = {a, e, i, o, u}, X is a set of vowels in the English alphabets.

Set builder form

In this method of representation, the set is described using the unique property shared by all of the elements of the set. Consider the set

Example 1:

A = {2, 3, 5, 7} What is the common property shared by the elements 2, 3, 5 and 7. They are all prime numbers less than 10. In set-builder form, also called the Rule method, we write it as A = {x/x is a prime number less than 10} expanded as “A is a set of elements x, such that x is a prime number less than 10”

Example 2:

Consider in Roster form the following set X = {3, 6, 9, 12, 15, 18}. In set-builder form, it is X = {x/x = 3n, where n ∈N and n < 5} n∈N stands for ‘n belongs to the set of natural numbers less than 5 the formula for set builder form is, therefore, S = {x/x has a property} or S = {x: x has a property}

SET THEORY CONCEPT NO. 3:

VARIOUS TYPES OF SETS

1. Various types of sets:

Finite set – A set that contains a limited number of elements is called a finite set.

Example 1.

A = {1, 3, 5, 7, 9}. Here A is a set of five positive odd numbers less than 10. Since the number of elements is limited, A is a finite set. A grade 5 class is a finite set, as the number of students is a fixed number.

Infinite set

A set that contains an unlimited number of elements is called an infinite set.

1. The set of natural numbers N, is an infinite set as the counting of numbers does not come to an end.

2. The set of integers is an infinite set.

Singleton set

A set that contains only one element is a singleton set.

Example 1:

A {set of even prime numbers}. Now A = {2}. The only even prime number is 2. All other prime numbers are odd. Therefore A can contain only one element, namely 2. Therefore A is a singleton set.

Example 2:

X = {x: x is neither positive nor negative}. Now, X = {0}, because it’s only 0 which is neither positive nor negative. Therefore, X is a singleton set.

Null set

A Set that does not contain any element is called an empty set or null set. S = {x: x ∈Z, x = 1/n, n ∈ N}, N is natural number and Z is an integer. Since n is an integer, 1/n cannot be an integer. Therefore, S cannot contain an element x which is an integer.

Note:

1. The Empty set is denoted as { } or by the Greek letter Φ

2. {{}} or {Φ} are not empty sets, because each contains one element, namely the empty set Φ itself.

Cardinal Number of a set or Cardinality of a set: The cardinality of a set is the number of elements a set contains. It is denoted as n (A). n (A) is read as the number of elements in set A

Example 1:

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

The cardinality of set A is 5. It is denoted as n (A) = 5

Note:

Example 1.

Let X = { }, then n (X) = 0

Let Z = {{}} or Z = {Φ}, then n (Y) = 1

Example 2:

Cardinality of infinite set is not defined.

Equivalent sets

Two sets that have the same number of elements, i.e. same cardinality are equivalent sets. P = {p. q. r, s, t} and Q = {a, e, i, o, u}. Since the two sets P and Q contain the same number of elements 5, therefore they are equivalent sets.

Equal sets

Two sets that contain the same elements are called equal sets. A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {

Overlapping sets

Two sets that have at least one common element are called overlapping sets.

Example 1:

X = {1, 2, 3} and Y = {3, 4, 5}. The two sets X and Y have an element 3 in common. Therefore they are called overlapping sets.

Example 2:

A = {x: x is an even prime number}, B = {x: x∈2n,n∈N }. The two sets A and B are overlapping sets because 2 is a common element in A and B.

Disjoint sets

C = {2, 4, 6} and D = {1, 3, 5}. The two sets C and D are disjoint sets as they do not have even one element in common.

Example 1:

E = {set of odd numbers} and F = {set of even numbers}. The two sets E and F are disjoint as no odd number is an even number nor any of even numbers is odd.

Subset

Set A is a subset of set B if every element of A is an element of set B. If set A is a subset of set B, then it is denoted as ACB

Example 1:

Let A = {1, 2, 3} and B = {2, 3, 4, 1}. Since every element of set A is present in set B too, we say A is a subset of B.

Example 2:

Let A = {multiples of 4} B = {multiples of 2}. Now, X = {1, 4, 8, 12…} and Y = {1, 2, 4, 6, 8, 10, 12…}. Since every element of set X is also an element of set Y, therefore. X is a subset of Y.

Note:

If two sets A and B are equal sets, then each one is a subset of the other. If A = {a, e, i, o, u} and B = {vowels of English alphabets}, then A = B. But, note that A C B, and BCA. Therefore, if A C B and BCA, then A = B. Every set is a subset of itself. ACA. The empty set is a subset of every set.

SET THEORY CONCEPT NO. 4:

OPERATIONS ON SETS

1. Union of Two Sets:

The Union of two sets A and B is an operation on the two sets. It is a set that contains all of the elements of both the sets A and B. The union of two sets A and B is denoted as AUB. Set builder notation for the union of two sets A and B is AUB = {x/x∈A or x∈B or x∈both A and B}

Example 1:

A = {1, 2, 3] and B = {2, 3, 4}. The Union of the two sets A and B is AUB = {1, 2, 3, 4}

Example 2:

X = {asia, america, australia} Y = {australia, europe, antarctica } Z = {europe, india, asia } X UYU Z = {asia, america, australia, europe, antarctica, india, u.s.a}

Note:

In the union of two sets, common elements need not be written more than one time. It is just enough to write common elements only once. In a set, an element is counted only once though it may be written any number of times. For example the set A = {1, 1, 1} contains only one element, 1, though 1 is written 3 times in the set.

2. Intersection of two sets:

The intersection of two sets A and B is the set which contains the elements that are in both A and B. It is denoted as A∩B. The intersection of two sets A and B in set-builder form is A∩B = { x/x∈A and x∈B}.

Example 1:

If A = {1, 2, 3} and B = {3, 4, 5}, then what is A∩B?

Answer:

A∩B = {1}

Example 2:

If A = {1, 2, 3} and B = {4, 5, 6}, then what is A∩B?

Answer:

Since there is no element which is in both the set A and set B, therefore
A∩B = Φ. We can also write A∩B = { }

Example 3:

A = {factors of 8} B = {factors of 9}. What is A∩B?

Answer:

A = {factors of 8} is the set {1, 2, 4, 8 …} and

B = {factors of 9} is the set {1, 3, 6, 9…}

So, we can see that 1 is the only element that is in both set A and set B.

The only element common to both sets A and B is 1. Therefore A∩B = {1}

3. Difference of two sets:

Let A and B be two sets. Then the difference of the two sets A and B is denoted as A – B. And, it is A – B = {x: x∈A but xB}. A – B is the set which contains elements of set A but not the elements of set B.

Example 1:

Let A = {1, 2, 3, 4} and B = {5, 6, 4, 3}. What is A – B?

Answer:

1 and 2 are the elements in set A but not in set B. Therefore, A – B = {1, 2}

Example 2:

Let A = {1, 2, 3, 4} and B = {5, 6, 4, 3}. What is B – A?

Answer:

5 and 6 are the two elements in set B which are not present in set A. Therefore, B – A = {5, 6}. From the above two examples, we can note that A – B ≠ B – A