"Study Set Theory with Ease Now"

 

SET THEORY
INTRODUCTION
In the English language, some nouns are called collective nouns because they refer to the collection of objects. You can study sets with all ease.

For example, army refers to a collection of soldiers, herd refers to a collection of animals. In mathematics, the
collection of objects refers to numbers, points, and ideas just to name a few.
Consider the following collections:
A = 1, 2 ,a, x, 6 , pencil, cup
B= John, pig, stone, goat, ink, Jane
C= 0, 4 , 5, 6, 8
D = rat, pig, cat, goat
We see that the collections in A and B are not well-defined as a common name cannot be given to each. Collections
C and D are be well-defined as a common name (noun) can be given to each. C is a collection of numbers and D is a
collection of animals. A is a collection of numbers, letters, pen , cup and so a common name cannot be given to A. B
is also a collection of persons, animals , and non-living objects, stone and ink.
DEFINITION OF A SET.
A set is a collection of well-defined objects. For example, the set of numbers; pens; students of a chosen height or
age; etc. We can know a set a soon as we know which subjects form it.
ELEMENTS OF A SET.
The objects or things that form a set are called elements or members of the set. We denote membership of a set
with Ꞓ and non-membership of the set with Ꞓ. We normally use capital letters to denote sets and small letters to
denote elements of the set. If x is an element of A and y is not an element of A, we write x Ꞓ A, y Ꞓ A .Elements of a
set are enclosed in curly brackets or braces {} .
Consider the set M= {0, 2, 4, 6.8.10}.
E is a set of even numbers between o and 10 inclusive. Ii follows that 2ꞒM, 6ꞒM. But 1 and 5 do not belong to M so
we write as 1ɆM, 5ɆM
EXAMPLE .Fill the spaces provided using Ꞓ or Ɇ
1) A ={4,8 12, 16}; 15 -----A
2) B={1,2,3,4,5}; 4 ------B
Solution
1) 15 does not belong to the set A. So you write: 15 ɆA 
2) 4 belongs to the B. So you write : 4ꞒB
EXERCISE
A) Fill the spaces provided below by using Ꞓ or Ɇ
i) P= { 11, 12 , 15 , 17 , 19, 20}; 4 --------P
ii) Q= { 10, 100 ,1000} ; 100 ----------Q
iii) E={ even numbers less than 8} ; 4 ------E
iv) K={ odd numbers less than 20} ; 23 -----K
v) V={ a,e,i,o,u} ; w -------V
B) State which of the following is/are True or False
i) 7 Ꞓ {Prime factors of 63}

ii) iii) iv) v)

Cod Ꞓ { fishes} 24 Ɇ { multiples of 5} Hexagon Ꞓ{quadrilaterals} Octagon Ɇ {polygons}

NAMING OF SETS
You can name a set in three ways namely
i) By writing a short statement describing its members.

ii) By listing the members of the set iii) By writing the characteristic property Example 1.
The set S of all square numbers less than 20 can be named as follows:
i)	S = {all square numbers less than 20}.

ii) iii)

S= {1, 4, 9, 16}. S={x/ x is a square number less than 20} and read as “S is a set of all x’s such that x is a square number less than 20”.

It should however be noted that some sets can only be named or defined by listing the elements just as other can be
named or defined by their characteristic property. For example A= {xꞒ N: x is prime}
EXAMPLE 2.
List the members of each of the following sets:
A= {odd numbers from 5 to 15 inclusive}
B = {even numbers less than 10}
SOLUTION
A= {5, 7 9 11 13, 15}
B= {2, 4, 6, 8, 10}
Example 3
Write down the characteristic property of each of the following sets:
M= {5, 10 15 20 25}
N = {brothers, sisters, father, mother}
SOLUTION
M ={x: x is a multiple of 5 less than 30}
N = {x: x is a family member}
EXAMPLE 4
List the elements of each of the following sets:
K= {x: x is a multiple of 4, 5 ˂ x ˂ 21}
P={x: x is a prime factor of 30}
SOLUTION
K= {8, 12, 16, 20}
P= {2, 5}

EXERCISE
1) List the members of the following sets:
D= {days of the week beginning with T}
P={ prime numbers less than 25}
2) Name the set C= {3, 5, 7, 11, 13, 17} using the characteristic property.

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