"Three Simple Ways to Describe Sets"

DESCRIBING THE MEMBERS OF A SET.
When we specify elements of a set, we are simply describing them. The most common
methods used to describe are:
 The verbal description method
 The listing or roster method
 The set builder notation method.

$Describing Sets$

a) THE VERBAL DESCRIPTION METHOD. The set is described in words using a verbal
statement. You have to ensure that the statement is well-defined.
EXAMPLES.

i)
ii)
iii)
iv)

A={ set of colors in a rainbow}
P={set of all prime numbers less than 10}
E={ set of even numbers between 23 and 37}
I={set of integers between -2 and -16}

b) THE LISTING METHOD OR ROSTER NOTATION. We refer to this method as the
roster notation because a roster is a list of elements, one after the other. This
method is also called the tabulation method. When using this method, we list the
elements of the set in a row between braces or curly brackets. You should always
separate elements using commas. This method is convenient for describing small
sets.
EXAMPLES
i) A= {2,3,5,7,11,13}

ii)
iii)
iv)
v)

V={ a,e, i,o,u}
E={ 2,4,6,8,10, 12}
C={white, blue, black, yellow}
I={-2,-4,-6,-8,-12,-14,-16}

vi) O={1,3,5,7,9,11,…}
ADVANTAGE OF THE LISTING METHOD.
The listing method is straightforward for describing small sets. Very easy to read and
write.
DISADVANTAGE OF THE LISTING METHOD.
It is only used for describing small sets. It cannot be used to describe very large and finite
sets.
c) SET BUILDER NOTATION. When using the set-builder notation, put a variable to
represent any element in the set. Add a brief description of a specific property
that is common to all members of that set. Make sure that the property you are
using to describe the elements of the set should be common to all elements of
that set. This will help you to tell clearly which elements belong to
the set and which ones do not. Below are some sets described using set-builder
notation
EXAMPLES

i)	E={x/x is an even number less than 17} or E={x: x is an even number less than 17}, where x is the variable.

ii) iii) iv)	M={ c/c is a color of the rainbow} or M={x: x is a color of the rainbow} N={y: y is a natural number less than 12} I={x/x is an integer between -1 and -17}
You can also use set builder-notation to describe intervals of real numbers shown
below: v) vi) vii)	D={x/ -2˂x˂5} P={x: 4≤x≤ 11} Q={x/3≤x˂5}

We read this as “set E is the set of all elements x, such that x is an even number
less than 17.” The slash (/) or colon (:) can be used interchangeably to replace the phrase
‘such that’ or ‘for which’ when describing the sets. You can use either the slash or colon to
separate the variable you have set for the property you are using to describe elements of
the set.
ADVANTAGES OF THE SET-BUILDER NOTATION METHOD

i)
ii)

It can be used to describe small sets.
It can be used to describe large finite and infinite sets

SUMMARY: DESCRIBING A SET USING ALL THREE METHODS
A= {set of all odd positive numbers less or equal to 5}
A= {1, 3, 5}
A={x: x is an odd number and 0<x<5}
In studying mathematics, you are going to meet different types of sets. These sets will be
described using the listing method or set builder-notation method. Numbers are subdivided
into the following sets namely:
i) Natural numbers (N)
ii) Whole numbers (W)
iii) Integers (Z)
iv) Rational numbers (Q)
v) Real Numbers (R)
vi) Complex Numbers ( C)

i)Natural numbers	N={x/x is natural number}	N={1,2,3,…}
ii)Whole numbers	W={x/x is a whole number}	W= {0, 1, 2,..}
iii )Integers	Z={x/x is an integer}	Z={…,-1,0,1,…}
iv)Rational Numbers	Q={x: x Ꞓp/q, q≠0}	Q={…,-⅟2,0,⅟2,}
v)Real numbers	R={x: x is a real number⧽

vi)Complex numbers

C={x: x is complex number⧽

C={x+ Yi /a, b ꞒR,
i is an
imaginary unit.⧽

Thus far, you have had so much fun describing sets. Now, its time to try out few
exercises to solve for practice.

$Describe Sets$

EXERCISES

1) Write down the set E the set of positive even numbers less than 10 using the
roster notation method.
SOLUTION
E={2, 4, 8,}
2) Describe set M containing all natural numbers less than 10 using:
i) Listing method

ii)Set-builder notation

SOLUTION

i)M={ 1,2,3,4,5,6,7,8,9}
ii)

M={x/x is a natural number less than 10}

3) Describe the set P where P={x: x ꞒR, 0˂x˂1} USING VERBAL description method.
SOLUTION
i) The set P of all real numbers greater than zero and less than 1.
4) Describe the set M where M= {1, 3, 5, 7, 9} using the set-builder notation method.
SOLUTION
i) M={x: x is an odd number between zero and ten}

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