Objectives: By the end of this subtopic learners should be able to:
Define a set.
Use symbols in sets correctly.
Use Venn diagrams to solve problems involving 2 or more sets.
LANGUAGE AND NOTATION
A.Concept of a Set
The concept of sets is used in everyday life situations. Some examples include soccer team, collection of books, collection of photographs and a group of pupils in a class. In mathematics and related subjects such a collection of objects is called a set.
B.Definition of a Set.
A set is a well or clearly defined collection, list or class of objects.
A set is defined by either listing of elements or describing the set i.e. describing that component that is common to the members.
A set is normally denoted by a capital letter, for example A, B, C…. whilst small letters may be used to denote members or elements of the set.
The members or elements of a set are enclosed in set braces {} and listed members are separated from the other by a semi-colon (;)
v) W = { Monday; Tuesday; Wednesday;Thursday;Friday;Saturday; Sunday}
W= {days of the week}
vi) Z = { lion; hyena; elephant; cheetah; buck}
Z= {wild animals}
N.B
It is noted that in each of the sets given above, the elements or members of the set have certain characteristics in common. However, this is not always a requirement, there can be a set whose elements have no relation other than being grouped together.
A.Set Builder Notation
Examples
1. N= {x: x is a natural number}
N= {1; 2; 3; 4…}
2.$P=\left\{x:-2\le x<20,andxisaninteger\right\}$
P = {-2;-1; 0; 1; 2; 3; 4…19}
B.Membership of a Set
If p is a member of a set M, that is, if M contains an element p as one of its elements then we write p ∈ M where ∈ is a symbol representing the term “member of”
If there is a letter q which does not belong to the set M, we write$q=\notin .$
Examples
Make each
of the following statement true by writing ∉. or ∈ in place of …..
(i) 7…..{ 2;3;5;7;9}
(ii)A….{ a; e, i; o; u}
(iii)February…… {days of the week}
(iv)-4….. {Integers}
Solutions
7∈ {2; 3;5; 7; 9}
(ii) A ∉{a; e; i;o; u}
(iii) February ∉ {days of the week}
(iv) -4 ∈ {integers}
c.Null or Empty Set.
Can a set have no elements in it at all?
The set with no elements in it, is called an empty set or a null set and is denoted by either {} or$\varnothing $
Examples
Determine whether or not the following are empty sets:
i) Perfect squares which are negative numbers.
ii) The set of perfect cubes amongst the numbers 1, 3, 5, 8 and 10.
iii) The set of vowels amongst the letters m, n, x, y, z.
Solution
i) Empty set because there are no negative numbers which are perfect squares. ii)Not empty because 1 and 8 are perfect cubes. iii)Empty because none of the letters is a vowel.
d.Finite Sets, Infinite Sets and Three Dots
If set V is the set of vowels {a; e; i; o;u}, then the number of elements in set V equals 5.
A finite set is a set with a definite number of elements, meaning V is a finite set.
Items which can be counted up to a certain number are said to be finite.
Number of elements in set V is denoted by n (V).
Therefore if V= {a; e; i, o; u} then, n(V) = 5
If N is the set of natural number then, N= {1; 2; 3; 4…} where the 3 dots denote that the set has an endless number of elements.
A set with an endless number of elements is infinite, meaning set N is infinite and there is no definite number of elements to it. n (N) cannot be determined.
An empty or null set has no elements in it. Therefore n$\left(\varnothing \right)$ = 0.
Three dots are used to mean and so on. They are used in putting down elements of an infinite set or a large finite set.
Finite set
A finite set has a number of elements which is a natural number that is, a definite positive whole number.
examples
1P={vowels}
P={a;e;i;o;u}
n(P)=5
2.
Infinite set and three dots
The number of elements in an infinite set is unlimited or uncountable by the natural numbers.
The number of elements in an infinite set could also be said to be infinite.
The elements cannot all be listed that is,endless.
Three dots are used to mean 'and so on'.
The three dots can also be used for finite sets which are too larger to have all elements listed.
Examples
A = {natural numbers}A = {1; 2; 3;4;5…}
B= {prime numbers} B = {2; 3; 5;7;11;13…}
e.Equality of Sets
Equality of sets means the sets contain exactly the same elements that is every element in one set is in the other set and vice versa.
If two sets M and N are equal, then we write M=N.
The order in which the elements of a set are listed is irrelevant as long as all the elements are listed between the braces. The two sets A= {1;2;3;4;} and B = {4;1;2;3}are equal i.e. A=B since each set contains four distinct elements 1, 2, 3 and 4.
Repetition of an element or elements of a set does not affect the set description.
The two sets P= {a, a, b, c, d, d, a} and Q= {a, b, c, d} are equal i.e. P=Q since the four distinct members contained in each set are a, b, c and d.
f.Disjoint Sets.
Two sets M and F are such that M is the set of male teachers in a school and F is the set of female staff members in the school.
The two sets are distinct because either one is male or female and cannot be both.
The two sets are disjoint as no member in M also belong to F.
Example
Suppose A= {6; 7}, B= {8 ;9} and C= {4; 5; 6}
Sets A and B are disjoint since they have no elements in common.
Sets A and C are not disjoint because they have a common element 6.
g.Universal/ Entire Set and Subsets
The universal set, often denoted by ε or Ω is the set containing all sets of interest under consideration.
Examples
1. If X= {1;2;3;4;5;6;7;8;9;10}, Y= {1;2;3;4;6}, Z= { 2;3;5;7}
All the elements of the sets Y and Z are contained in X.
X, therefore, acts as the universal set.
All the members of the sets Y and Z belong to set X and each one of them is a subset of X.
2. Y is a subset of set X is denoted by Y $\subseteq $ X
3.Proper Subset
Given set V= {a; e; i;o;u} then it subsets are;
{a},{e}, {i}, {o}, {u},
{a;e},{a;i},{a;o},{a;u},{e;i},{e;o},{e;u},{i;o},{i;u},{o;u},
{a;e;i},{a;e;o},{a;e;u},{a;i;o},{a;i;u},{a;o;u},{e;i;o},{e;i;u},{e;o;u},{i;o;u}
{a;e;i;o},{a;e;i;u},{a; i;o ;u},{e; i; o;u};
{a; e; i;o; u}, {}.
The set V has 16 subsets, with the first 14 having elements which are part of set V and the last two equal to V and taking nothing from V respectively.
The subsets which contain part of V are said to be proper subsets.
{a, e} is a proper subset of set V, is denoted by ;
{a, e} $\subset $V.
N.B
A set can be a subset of itself and an empty set is a subset of every set.
Suppose at Mukai College there are three categories of teachers, the College Diploma holders, the Certificate holders and the Degree holders.
The universal set is of all teachers at Mukai College.
The complement of Degree holders at the college is the set of all teachers at the college who are not degree holders.
Thus the complement of a given set includes all members in the universal set that do not belong to that set.
Complement of set A refers to things not in (that is, things outside of) A.
If A is a subset of B then the complement of set A is the set of elements in B but not in A.
If ξ be the universal set, and A is the subset of ξ, then the complement of set A is the set of all elements of ξ which are not the element of A.
Complement of a set is a set of elements not in the set but in the universal set.
Example
If ξ = { 1;2;3,…12}, A= {2;3;4,5;6}, B= {2;4;6;8;10} find,
(a) A’
(b) B’
(c) ξ’
(d){ }’
Answer.
(a)A’ = {1; 7;8;9;10;11;12}
(b)B’ = {1; 3; 5;7; 9; 12}
(c) ξ’= { }
(d){ }’= ξ
i.INTERSECTION OF SETS
The intersection of two set A and B is the set of elements common to both the sets.
It is the largest set which contains all the elements that are common to both the sets.
The symbol for denoting intersection of set is $\text{'}\cap \text{'}$
Examples
1.Let set X = {2,3,4,5,6} and set Y = { 3,5,7,9}
In the two sets, the elements 3 and 6 are common. The set containing these common elements, that is, {3,5} is the intersection of X and Y.
Symbolically the intersection of the two sets X and Y is written as X ∩ Y.
The set which is the intersection of the two set X and Y can be represented as X∩ Y = { x: x ∈X and x∈Y }
2.If A={ 2,4,6,8,10} and B= { 1,3,8,4,6},
Find the intersection of A and B.
Solution
A ∩B= {4; 6; 8}
3.If X= {a;b;c} and Y=∅, find the intersection of X and ∅.
Solution
X ∩∅ = ∅
4. If set A = {4; 6,8;10;12}, set B = {3;6; 9;12;15;18} and
set C = {1;2;3;4;5;6;7;8;9;10}, find;
(i) A ∩ B
(ii) B ∩C
(iii)A ∩ C
(iv)A ∩B∩C
Solution
(i)A ∩ B = {6;2}
(ii)B ∩ C = {3;6;9}
(iii)A ∩ C= {4;6;8;10}
(iv)A ∩B∩C= {6}
N.B
A ∩ B = B ∩ C
A ∩ B is a subset of sets A and set B
∅∩ A = ∅
ξ ∩ A = A
j.Union of Sets
Given at least two sets, the union is the set that contains elements or objects that belong to A or to B or to both.
Common elements are not repeated in the combination.
The union of set A and set B is written as A ∪B.
Examples;
1.If set A={1;2;4; 6}, and set B= {4; a;b;c;d;f} then,
AUB = {1; 2; 4; 6;a; b;c; d; f}
2. If X= {x: x is a number bigger than 4 and smaller than 8}.
Y= {x: x is a positive number smaller than 7}
X= {5; 6;7}, Y = {1; 2; 3; 4; 5; 6}
X∪ Y= { 1;2;3;4;5;6;7} or X ∪ Y = { x: x is a number bigger than 0 and smaller than 8}.
3. If set P= {#; %;*;!;$} and Q = {} then,
P∪Q = {#;%;*;!;$}
= set P.
n (P ∪Q) = 5
4. If ξ = {a; b;c;d;e;f; g} and set M= {b;d;g} then
ξ ∪ M ={ a;b;c;d;e;f;g} = ξ, where ξ is the universal set.
n(ξ∪M)=7.
k.Complements of Intersections and Unions
Intersections and Unions are also subsets of the sets they were derived from and can also have complements.
Examples
1.If ξ = {p;q;r;s;t;u}, set D = {p;r; t;u} and R= {q;s;t}, list the members of the following sets.
(a)D'
(b)R'
(c)D'∪ R
(d)D ∪ R'
(e)D'∩R
(f)D' ∪D
(g)R'∩R
(h)(D∪R)'
(i)(D∩R)'
Solution;
(a)D' ={ q;s}
(b)R' = { p;r;u}
(c)D'∪R = { q;s;t }
(d)D ∪ R' = { p;r;t;}
(e)D'∩ R = { q; s}
(f)D' ∪ D = {q;s;p;r; t ;u} = ξ.
(g)R' ∩ R = { }or ∅
(h)(D∪ R)' = {}
(i)( D ∩ R)' = { p;q;r;s;u}
l.Set Notation/ Symbols
The table below shows symbols used in Sets and their meaning:
Symbols
Meaning
x= { p, q, r, s}
X is the set p, q, r, s
N= { 1,2,3,4,5 …}
… three dots mean ‘ and so on’.
P= { x: x is an even number}
Q = { (x ; y) : y= 2x+ 5}
R = { x: a < x ≤ b}
P, Q and R are examples of set- builder notation.
Sets P and R give possible values of x and set Q is the set of point with co-ordinates (x:y)
∅ or { }
The empty set or null set.
ξ or U
The universal set.
∈
Is a member of
∉
Is not a subset of
n(P)
Number of elements in set P
If X = { a, b, c, d}, then n(X) = 4
A'
Compliment of set A.
X$\subset $Y
X is a proper subset of Y or
X is contained in Y
Y$\supset $X
Y contains X
X⊆X ∅⊆A
X is a subset of X
∅ is a subset of A
XUP
Union of X and P
X∩P
Intersection of X and P
m.Venn Diagrams
Venn diagrams are used to display the elements or numbers of elements in sets and subsets.
Venn diagrams also help to show relationships between sets, for example, unions, intersection and complements.
Examples
1.ξ = {1;2;3;4… 11;12}
A = {2;3;4;5;6}
B = {2;4;6;8;10}
(a) A∩B = {2;4;6}.
(b) B’ = {1;3;5;7;9;11;12}
(c) A∪B = {2;3;4;5;6;8;10}
(d) n(A∪B)’ =5
The Venn diagram shows the universal set E and the
Subsets X,Y and Z.
The letters in the regions of the Venn diagram represent
The numbers of elements in each set or subset.
The Venn diagram shows the universal set E and the
Subsets X,Y and Z.
The letters in the regions of the Venn diagram represent
The numbers of elements in each set or subset.
Examples
1
(i) n(Z) = a+b+c+d
=4
(ii)n(X∩Y∩Z) = a
(iii)n(X∪Y∪Z) = g
2.The diagrams below shows the universal set E and the subsets A and B.
In each diagram shade the region named below it.