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A = { x: x is a vowel in English alphabet }

B = { x: x is an integer and 0 < x < 10 }

The roaster equivalents of two sets are :

A = { a , e , i , o , u }

B = { 1,2,3,4,5,6,7,8,9 }

Can we write set “B” as the one which comprises single digit natural number? Yes. Thus, we can see that there are indeed different ways to define and identify members and hence the flexibility in defining collection.

We should be careful in using words like “and” and “or” in writing qualification for the set. Consider the example here :

C = { x: x Z and 2 < x < 4 }

Both conditional qualifications are used to determine the collection. The elements of the set as defined above are integers. Thus, the only member of the set is “3”.

Now, let us consider an example, which involves “or” in the qualification,

C = { x: x A or x B }

The member of this set can be elements belonging to either of two sets "A" and "B". The set consists of elements (i) belonging exclusively to set "A", (ii) elements belonging exclusively to set "B" and (iii) elements common to sets "A" and "B".

Example

Problem 1 : A set in roaster form is given as :

A = { 5 2 6 , 6 2 7 , 7 2 8 }

Write the set in “set builder form”.

Solution : We see here that we are dealing with natural numbers. The numerators are square of natural numbers in sequence. The number in denominator is one more than numerator for each member. We can denote natural number by “n”. Clearly, if numerator is “ n 2 ”, then denominator is “n+1”. Therefore, the expression that represent a member of the set is :

x = n 2 n + 1

However, this set is not an infinite set. It has exactly three members. Therefore, we need to specify “n” so that only members of the set are exclusively denoted by the above expression. We see here that “n” is greater than 4, but “n” is less than 8. For representing three elements of the set,

5 n 7

We can write the set, now, in the builder form as :

A = { x : x = n 2 n + 1 , where "n" is a natural number and 5 n 7 }

In set builder form, the sequence within the range is implied. It means that we start with the first valid natural number and proceed sequentially till the last valid natural number.

Some important sets representing numbers

Few key number sets are used regularly in mathematical context. As we use these sets often, it is convenient to have predefined symbols :

  • P(prime numbers)
  • N (natural numbers)
  • Z (integers)
  • Q (rational numbers)
  • R (real numbers)

We put a superscript “+”, if we want to specify membership of only positive numbers, where appropriate. " Z + ", for example, means set of positive integers.

Empty set

An empty set has no member or object. It is denoted by symbol “φ” and is represented by a pair of braces without any member or object.

φ = { }

The empty set is also called “null” or “void” set. For example, consider a definition : “the set of integer between 1 and 2”. There is no integer within this range. Hence, the set corresponding to this definition is an empty set. Consider another example :

B = { x : x 2 = 4 and x is odd }

An odd integer squared can not be even. Hence, set “B” also does not have any element in it.

There is a bit of paradox here. If the definition does not yield an element, then the set is not well defined. We may be tempted to say that empty set is not a set in the first place. However, there is a practical reason to have an empty set. It enables mathematical operations. We shall find many examples as we study operations on sets.

Equal sets

The members of two equal sets are exactly same. There is nothing more to it. However, we need to know two special aspects of this equality. We mentioned about repetition of elements in a set. We observed that repetition of elements does not change the set. Consider example here :

A = { 1,5,5,8,7 } = { 1,5, 8,7 }

Another point is that sequence does not change the set. Therefore,

A = { 1,5,8,7 } = { 5,7,8,1 }

In the nutshell, when we have to compare two sets we look for distinct elements only. If they are same, then two sets in question are equal.

Cardinality

Cardinality is the numbers of elements in a set. It is denoted by modulus of set like |A|.

Cardinality
The cardinality of a set “A” is equal to numbers of elements in the set.

The cardinality of an empty set is zero. The cardinality of a finite set is some positive integers. The cardinality of a number system like integers is infinity. Curiously, the cardinality of some infinite set can be compared. For example, the cardinality of natural numbers is less than that of integers. However, we can not make such deduction for the most case of infinite sets.

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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