Download Sets and Venn diagrams

3
Chapter
Sets and
Venn diagrams
Syllabus reference: 2.1, 3.1, 3.2
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Contents:
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\095IB_STSL-2_03.CDR Friday, 26 March 2010 12:10:49 PM PETER
Set notation
Special number sets
Set builder notation
Complements of sets
Venn diagrams
Venn diagram regions
Numbers in regions
Problem solving with
Venn diagrams
IB_STSL-2ed
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SETS AND VENN DIAGRAMS
(Chapter 3)
OPENING PROBLEM
A city has three football teams A, B and C, in the national league. In the
last season, 20% of the city’s population went to see team A play, 24% saw
team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw
both A and C, and 6% saw both B and C. 1% saw all three teams play.
Things to think about:
a How can we represent this information on a diagram?
b What percentage of the population:
i saw only team A play
ii saw team A or team B play but not team C
iii did not see any of the teams play?
A
SET NOTATION
A set is a collection of numbers or objects.
For example, the set of all digits we use to write numbers is f0, 1, 2, 3, 4, 5, 6, 7, 8, 9g.
Notice how the ten digits have been written within the brackets “f” and “g”, and how they
are separated by commas.
We usually use capital letters to represent sets, so we can refer to the set easily.
For example, if V is the set of all vowels, then V = fa, e, i, o, ug.
Every number or object in a set is called an element or member of the set.
We use the symbol 2 to mean is an element of and 2
= to mean is not an element of.
So, for the set A = f1, 2, 3, 4, 5, 6, 7g we can say 4 2 A but 9 2
= A.
The set f g or ? is called the empty set and contains no elements.
COUNTING ELEMENTS OF SETS
The number of elements in set A is written n(A).
For example, the set A = f 2, 3, 5, 8, 13, 21g has 6 elements, so we write n(A) = 6.
1st 2nd 3rd
A set which has a finite number of elements is said to be a finite set.
For example: A = f2, 3, 5, 8, 13, 21g is a finite set.
? is also a finite set, since n(?) = 0.
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Infinite sets are sets which have infinitely many elements.
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SETS AND VENN DIAGRAMS
(Chapter 3)
97
For example, the set of counting numbers 1, 2, 3, 4, .... does not have a largest element,
but rather keeps on going forever. It is therefore an infinite set.
SUBSETS
Suppose P and Q are two sets. P is a subset of Q if every element
of P is also an element of Q. We write P µ Q:
For example, f2, 3, 5g µ f1, 2, 3, 4, 5, 6g as every element in the first set is also in the
second set.
We say P is a proper subset of Q if P is a subset of Q but is not equal to Q.
We write P ½ Q.
UNION AND INTERSECTION
If P and Q are two sets then:
² P \ Q is the intersection of P and Q and consists
of all elements which are in both P and Q.
² P [ Q is the union of P and Q and consists of all
elements which are in P or Q.
For example, if P = f1, 3, 4g and Q = f2, 3, 5g then
Every element in P and
every element in Q is
found in P [ Q.
DEMO
P \ Q = f3g and P [ Q = f1, 2, 3, 4, 5g.
DISJOINT SETS
Two sets are disjoint or mutually exclusive if they have no elements in common.
Example 1
Self Tutor
M = f2, 3, 5, 7, 8, 9g and N = f3, 4, 6, 9, 10g
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i No. Not every element of M is an element of N .
ii Yes, as 9, 6 and 3 are also in N .
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i M \ N = f3, 9g since 3 and 9 are elements of
both sets.
ii Every element which is in either M or N is in the
union of M and N .
) M [ N = f2, 3, 4, 5, 6, 7, 8, 9, 10g
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i 4 is not an element of M, so 4 2 M is false.
ii 6 is not an element of M, so 6 2
= M is true.
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a True or false?
i 42M
ii 6 2
=M
b List the sets:
i M \ N ii M [ N
c Is
i M µN
ii f9, 6, 3g µ N ?
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\097IB_STSL-2_03.CDR Monday, 14 December 2009 9:48:24 AM PETER
To write down M [ N,
start with M and add to
it the elements of N
which are not in M.
IB_STSL-2ed
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SETS AND VENN DIAGRAMS
(Chapter 3)
EXERCISE 3A
1 Write using set notation:
a 5 is an element of set D
b 6 is not an element of set G
c d is not an element of the set of all English vowels
d f2, 5g is a subset of f1, 2, 3, 4, 5, 6g
e f3, 8, 6g is not a subset of f1, 2, 3, 4, 5, 6g.
2 Find i A \ B ii A [ B for:
a A = f6, 7, 9, 11, 12g and B = f5, 8, 10, 13, 9g
b A = f1, 2, 3, 4g and B = f5, 6, 7, 8g
c A = f1, 3, 5, 7g and B = f1, 2, 3, 4, 5, 6, 7, 8, 9g
3 Write down the number of elements in the following sets:
a A = f0, 3, 5, 8, 14g
b B = f1, 4, 5, 8, 11, 13g
c A\B
d A[B
4 Describe the following sets as either finite or infinite:
a the set of counting numbers between 10 and 20
b the set of counting numbers greater than 5.
5 Which of these pairs of sets are disjoint?
a A = f3, 5, 7, 9g and B = f2, 4, 6, 8g
b P = f3, 5, 6, 7, 8, 10g and Q = f4, 9, 10g
6 True or false? If R and S are two non-empty sets and R \ S = ? then R and S are
disjoint.
B
SPECIAL NUMBER SETS
The following is a list of some special number sets you should be familiar with:
N = f0, 1, 2, 3, 4, 5, 6, 7, ....g is the set of all natural or counting numbers.
Z = f0, §1, §2, §3, §4, ....g is the set of all integers.
Z + = f1, 2, 3, 4, 5, 6, 7, ....g is the set of all positive integers.
Z ¡ = f¡1, ¡2, ¡3, ¡4, ¡5, ....g is the set of all negative integers.
Q is the set of all rational numbers, or numbers which can be written in the form
p
q
where p and q are integers and q 6= 0.
R is the set of all real numbers, which are all numbers which can be placed on the number
line.
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\098IB_STSL-2_03.CDR Wednesday, 10 February 2010 3:55:10 PM PETER
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SETS AND VENN DIAGRAMS
(Chapter 3)
99
Notice that the set of integers is made up of the set of negative integers, zero, and the set of
positive integers.
Z = (Z ¡ [ f0g [ Z + )
The sets N , Z , Z + , Z ¡ , Q and R are all infinite sets.
Example 2
Self Tutor
a any positive integer is also a rational number
b ¡7 is a rational number
Explain why:
a We can write any positive integer as a fraction where the number itself is the
numerator, and the denominator is 1.
For example, 5 = 51 .
So, all positive integers are rational numbers.
b ¡7 =
¡7
1 ,
so ¡7 is rational.
Example 3
Self Tutor
All terminating
decimal numbers are
rational numbers.
Show that the following are rational numbers:
a 0:47
b 0:135
47
100 ,
135
= 1000
a 0:47 =
b 0:135
so 0:47 is rational.
=
27
200 ,
so 0:135 is rational.
EXERCISE 3B.1
1 Show that 8 and ¡11 are rational numbers.
2 Why is
4
0
not a rational number?
3 True or false?
a Z+ µN
e Q ½Z
b N ½Z
c N =Z+
d Z¡ µZ
f f0g µ Z
g Z µQ
h Z+[Z¡ =Z
4 Show that the following are rational numbers:
a 0:8
b 0:71
5 True or false?
a 127 2 N
b
c 0:45
138
279
d 0:219
c 3 17 2
=Q
2Q
e 0:864
4
d ¡ 11
2Q
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6 Describing the following sets as either finite or infinite:
a the set of all rational numbers Q
b the set of all rational numbers between 0 and 1.
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100
SETS AND VENN DIAGRAMS
(Chapter 3)
RECURRING DECIMALS (EXTENSION)
All recurring decimal numbers can be shown to be rational.
Example 4
Self Tutor
a 0:777 777 7::::
Show that:
a
b 0:363 636 ::::
Let x = 0:777 777 7::::
) 10x = 7:777 777 7::::
) 10x = 7 + 0:777 777 ::::
) 10x = 7 + x
) 9x = 7
) x = 79
are rational.
Let x = 0:363 636 ::::
) 100x = 36:363 636 ::::
) 100x = 36 + 0:363 636 ::::
) 100x = 36 + x
) 99x = 36
) x = 36
99
4
) x = 11
b
So, 0:777 777:::: = 79 ,
which is rational.
So, 0:363 636 :::: =
which is rational.
7 Show that these are rational:
a 0:444 444 ::::
b 0:212 121 ::::
4
11 ,
c 0:325 325 325 ::::
IRRATIONAL NUMBERS
All real numbers are either rational or irrational.
Irrational numbers cannot be written in the form
p
q
where p and q are integers, q 6= 0.
The set of irrational numbers is denoted by Q 0 .
Numbers such as
p
p p p
2, 3, 5, and 7 are all irrational.
This means that their decimal expansions neither terminate nor recur.
Other irrationals include ¼ ¼ 3:141 593 .... which is the ratio of a circle’s circumference to
its diameter, and exponential e ¼ 2:718 281 828 235 :::: which has applications in finance
and modelling.
The following shows the relationship between various number sets:
Z+
positive integers
Z¡
negative integers
f0g
zero
Q
rationals
R
reals
Z
integers
Q [Q 0 =R
Q \Q 0 =?
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Z + [ f0g [ Z ¡ = Z
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Q 0
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\100IB_STSL-2_03.CDR Friday, 26 March 2010 3:58:42 PM PETER
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SETS AND VENN DIAGRAMS
(Chapter 3)
101
EXERCISE 3B.2
1 Which of the following are irrational numbers?
p
p
8
c
4
a 3:127
b
q
9
2 Show that
25 is a rational number.
3 On the table, indicate with a tick or cross
whether the numbers in the left hand column
belong to Q , R , Z , Q 0 or N .
d
p
2
Q
R
p
1
Z
Q
0
N
5
¡ 13
2:17
¡9
C
SET BUILDER NOTATION
A = fx j ¡2 6 x 6 4, x 2 Z g reads “the set of all x such that x is an integer between
¡2 and 4, including ¡2 and 4.”
such that
the set of all
We can represent A on a number line as:
x
-2 -1
0
1
2
3
4
We see that A is a finite set, and that n(A) = 7.
B = fx j ¡2 6 x < 4, x 2 R g reads “the set of all real x such that x is greater than or
equal to ¡2 and less than 4.”
We represent B on a number line as:
filled in circle
indicates -2 is
included
open circle
indicates 4 is
not included
x
B is an infinite set, and n(B) = 1:
-2
Example 5
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Suppose A = fx j 3 < x 6 10, x 2 Z g.
a Write down the meaning of the set builder notation.
b List the elements of set A.
c Find n(A).
a The set of all x such that x is an integer between 3 and 10, including 10.
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c There are 7 elements, so n(A) = 7.
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b A = f4, 5, 6, 7, 8, 9, 10g
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\101IB_STSL-2_03.CDR Monday, 14 December 2009 9:48:35 AM PETER
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102
SETS AND VENN DIAGRAMS
(Chapter 3)
EXERCISE 3C
1 Explain whether the following sets are finite or infinite:
a fx j ¡2 6 x 6 1, x 2 Z g
b fx j ¡2 6 x 6 1, x 2 R g
c fx j x > 5, x 2 Z g
d fx j 0 6 x 6 1, x 2 Q g
2 For the following sets:
i Write down the meaning of the set builder notation.
ii If possible, list the elements of A.
iii Find n(A).
a A = fx j ¡1 6 x 6 7, x 2 Z g
b A = fx j ¡2 < x < 8, x 2 N g
c A = fx j 0 6 x 6 1, x 2 R g
d A = fx j 5 6 x 6 6, x 2 Q g
3 Write in set builder notation:
a the set of all integers between ¡100 and 100
b the set of all real numbers greater than 1000
c the set of all rational numbers between 2 and 3, including 2 and 3.
4 fag has two subsets, ? and fag. fa, bg has four subsets: ?, fag, fbg, and fa, bg.
a List the subsets of i fa, b, cg ii fa, b, c, dg and state the number of subsets
in each case.
b Copy and complete: “If a set has n elements then it has ...... subsets.”
5 Is A µ B in these cases?
a A = ? and B = f2, 5, 7, 9g
b A = f2, 5, 8, 9g and B = f8, 9g
c A = fx j 2 6 x 6 3, x 2 R g and B = fx j x 2 R g
d A = fx j 3 6 x 6 9, x 2 Q g and B = fx j 0 6 x 6 10, x 2 R g
e A = fx j ¡10 6 x 6 10, x 2 Z g and B = fz j 0 6 z 6 5, z 2 Z g
f A = fx j 0 6 x 6 1, x 2 Q g and B = fy j 0 < y 6 2, y 2 Q g
D
COMPLEMENTS OF SETS
UNIVERSAL SETS
Suppose we are only interested in the natural numbers from 1 to 20, and we want to consider
subsets of this set. We say the set U = fx j 1 6 x 6 20, x 2 N g is the universal set in
this situation.
The symbol U is used to represent the universal set under consideration.
COMPLEMENTARY SETS
If the universal set U = f1, 2, 3, 4, 5, 6, 7, 8g, and the set A = f1, 3, 5, 7, 8g, then the
complement of A is A0 = f2, 4, 6g.
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The complement of A, denoted A0 , is the set of all elements of U which are not in A.
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SETS AND VENN DIAGRAMS
(Chapter 3)
103
Three obvious relationships are observed connecting A and A0 . These are:
² A \ A0 = ?
² A [ A0 = U
² n(A) + n(A0 ) = n(U )
as A0 and A have no common members.
as all elements of A and A0 combined make up U.
Example 6
Self Tutor
Find C 0 given that:
a U = fall positive integersg and C = fall even integersg
b C = fx j x > 2, x 2 Z g and U = Z
a C 0 = fall odd integersg
b C 0 = fx j x 6 1, x 2 Z g
Example 7
Self Tutor
If U = fx j ¡5 6 x 6 5, x 2 Z g, A = fx j 1 6 x 6 4, x 2 Z g and
B = fx j ¡3 6 x < 2, x 2 Z g, list the elements of these sets:
a A
e A\B
c A0
g A0 \ B
b B
f A[B
a A = f1, 2, 3, 4g
d B0
h A0 [ B 0
b B = f¡3, ¡2, ¡1, 0, 1g
0
c A = f¡5, ¡4, ¡3, ¡2, ¡1, 0, 5g
d B 0 = f¡5, ¡4, 2, 3, 4, 5g
e A \ B = f1g
f A [ B = f¡3, ¡2, ¡1, 0, 1, 2, 3, 4g
g A0 \ B = f¡3, ¡2, ¡1, 0g
h A0 [ B 0 = f¡5, ¡4, ¡3, ¡2, ¡1, 0, 2, 3, 4, 5g
EXERCISE 3D
1 Find the complement of C given that:
a U = fletters of the English alphabetg and C = fvowelsg
b U = fintegersg and C = fnegative integersg
c U = Z and C = fx j x 6 ¡5, x 2 Z g
d U = Q and C = fx j x 6 2 or x > 8, x 2 Q g
2 If U = fx j 0 6 x 6 8, x 2 Z g, A = fx j 2 6 x 6 7, x 2 Z g and
B = fx j 5 6 x 6 8, x 2 Z g list the elements of:
b A0
f A[B
a A
e A\B
c B
g A \ B0
d B0
3 If n(U ) = 15, n(P ) = 6 and n(Q0 ) = 4 where P and Q0 are subsets of U , find:
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\103IB_STSL-2_03.CDR Friday, 8 January 2010 9:15:17 AM PETER
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104
SETS AND VENN DIAGRAMS (Chapter 3)
4 True or false?
a If n(U ) = a and n(A) = b then n(A0 ) = b ¡ a.
b If Q is a subset of U then Q0 = fx j x 2
= Q, x 2 U g.
5 If U = fx j 0 < x 6 12, x 2 Z g, A = fx j 2 6 x 6 7, x 2 Z g,
B = fx j 3 6 x 6 9, x 2 Z g, and C = fx j 5 6 x 6 11, x 2 Z g,
list the elements of:
a B0
b C0
c A0
d A\B
0
0
0
e (A \ B)
f A \C
g B [C
h (A [ C) \ B 0
Example 8
Self Tutor
Suppose U = fpositive integersg, P = fmultiples of 4 less than 50g and
Q = fmultiples of 6 less than 50g.
b Find P \ Q.
a List P and Q.
c Find P [ Q.
d Verify that n(P [ Q) = n(P ) + n(Q) ¡ n(P \ Q).
a P = f4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48g
Q = f6, 12, 18, 24, 30, 36, 42, 48g
b P \ Q = f12, 24, 36, 48g
c P [ Q = f4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48g
d n(P [ Q) = 16 and n(P ) + n(Q) ¡ n(P \ Q) = 12 + 8 ¡ 4 = 16
So, n(P [ Q) = n(P ) + n(Q) ¡ n(P \ Q) is verified.
6 Suppose U = Z + , P = fprime numbers < 25g, and Q = f2, 4, 5, 11, 12, 15g.
b Find P \ Q.
a List P .
c Find P [ Q.
d Verify that n(P [ Q) = n(P ) + n(Q) ¡ n(P \ Q):
7 Suppose U = Z + , P = ffactors of 30g, and Q = ffactors of 40g.
b Find P \ Q.
a List P and Q.
c Find P [ Q.
d Verify that n(P [ Q) = n(P ) + n(Q) ¡ n(P \ Q).
8 Suppose U = Z + , M = fmultiples of 4 between 30 and 60g, and
N = fmultiples of 6 between 30 and 60g.
a List M and N .
b Find M \ N.
c Find M [ N .
d Verify that n(M [ N ) = n(M ) + n(N) ¡ n(M \ N ).
9 Suppose U = Z , R = fx j ¡2 6 x 6 4, x 2 Z g, and S = fx j 0 6 x < 7, x 2 Z g.
a List R and S.
b Find R \ S.
d Verify that n(R [ S) = n(R) + n(S) ¡ n(R \ S).
c Find R [ S.
10 Suppose U = Z , C = fy j ¡4 6 y 6 ¡1, y 2 Z g, and D = fy j ¡7 6 y < 0, y 2 Z g.
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c Find C [ D.
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a List C and D.
b Find C \ D.
d Verify that n(C [ D) = n(C) + n(D) ¡ n(C \ D).
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\104IB_STSL-2_03.CDR Thursday, 11 February 2010 10:10:42 AM PETER
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SETS AND VENN DIAGRAMS (Chapter 3)
105
11 Suppose U = Z + , P = ffactors of 12g, Q = ffactors of 18g, and R = ffactors of 27g.
a List the sets P , Q, and R.
b Find:
i P \Q
iv P [ Q
c Find:
i P \Q\R
ii P \ R
v P [R
ii P [ Q [ R
iii Q \ R
vi Q [ R
12 Suppose U = Z + , A = fmultiples of 4 less than 40g,
B = fmultiples of 6 less than 40g, and C = fmultiples of 12 less than 40g.
a
b
c
d
List the sets A, B and C.
Find:
i A\B
ii B \ C
iii A \ C
iv A \ B \ C
Find A [ B [ C.
Verify that n(A [ B [ C) = n(A) + n(B) + n(C) ¡ n(A \ B) ¡ n(B \ C)
¡ n(A \ C) + n(A \ B \ C):
13 Suppose U = Z + , A = fmultiples of 6 less than 31g,
B = ffactors of 30g, and C = fprimes < 30g.
a
b
c
d
List the sets A, B and C.
Find:
i A\B
ii B \ C
iii A \ C
iv A \ B \ C
Find A [ B [ C.
Verify that n(A [ B [ C) = n(A) + n(B) + n(C) ¡ n(A \ B) ¡ n(B \ C)
¡ n(A \ C) + n(A \ B \ C).
E
VENN DIAGRAMS
Venn diagrams are often used to represent sets of objects, numbers, or things.
A Venn diagram consists of a universal set U represented by a rectangle.
Sets within the universal set are usually represented by circles.
For example:
is a Venn diagram which shows set A within
the universal set U .
A
A0 , the complement of A, is the shaded region
outside the circle.
A'
U
Suppose U = f2, 3, 5, 7, 8g, A = f2, 7, 8g and A0 = f3, 5g.
We can represent these sets by:
A
5
2
7
A'
8
3
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106
SETS AND VENN DIAGRAMS
(Chapter 3)
SUBSETS
If B µ A then every element of B is also in A.
BµA
The circle representing B is placed within the circle
representing A.
B
A
U
INTERSECTION
A\B
A \ B consists of all elements common to both A
and B.
A
It is the shaded region where the circles representing
A and B overlap.
B
U
UNION
A[B
A
A [ B consists of all elements in A or B or both.
It is the shaded region which includes everywhere in
either circle.
B
U
DISJOINT OR MUTUALLY EXCLUSIVE SETS
Disjoint sets do not have common elements.
They are represented by non-overlapping circles.
U
For example, if A = f2, 3, 8g and B = f4, 5, 9g
then A \ B = ?.
A
B
Example 9
Self Tutor
Given U = f1, 2, 3, 4, 5, 6, 7, 8g, illustrate on a Venn diagram the sets:
a A = f1, 3, 6, 8g and B = f2, 3, 4, 5, 8g
b A = f1, 3, 6, 7, 8g and B = f3, 6, 8g.
a A \ B = f3, 8g
3
8
6
B
2
2
4
7
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4
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b A \ B = f3, 6, 8g, B µ A
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SETS AND VENN DIAGRAMS
Example 10
Self Tutor
Given U = f1, 2, 3, 4, 5, 6, 7, 8, 9g, illustrate on a Venn
diagram the sets A = f2, 4, 8g and B = f1, 3, 5, 9g.
(Chapter 3)
107
Since A and B are
disjoint, their circles
are separated.
A\B =?
A
7
2
B
3
1
4
8
5
9
6
U
EXERCISE 3E
1 Represent sets A and B on a Venn diagram, given:
a U = f2, 3, 4, 5, 6, 7g, A = f2, 4, 6g, and B = f5, 7g
b U = f2, 3, 4, 5, 6, 7g, A = f2, 4, 6g, and B = f3, 5, 7g
c U = f1, 2, 3, 4, 5, 6, 7g, A = f2, 4, 5, 6g, and B = f1, 4, 6, 7g
d U = f3, 4, 5, 7g, A = f3, 4, 5, 7g, and B = f3, 5g
2 Suppose U = fx j 1 6 x 6 10, x 2 Z g, A = fodd numbers < 10g, and
B = fprimes < 10g.
a List sets A and B.
b Find A \ B and A [ B.
c Represent the sets A and B on a Venn diagram.
3 Suppose U = fx j 1 6 x 6 9, x 2 Z g, A = ffactors of 6g, and B = ffactors of 9g.
a List sets A and B.
b Find A \ B and A [ B.
c Represent the sets A and B on a Venn diagram.
4 Suppose U = feven numbers between 0 and 30g,
P = fmultiples of 4 less than 30g, and
Q = fmultiples of 6 less than 30g.
a List sets P and Q.
b Find P \ Q and P [ Q.
c Represent the sets P and Q on a Venn diagram.
5 Suppose U = fx j x 6 30, x 2 Z + g, R = fprimes less than 30g, and
S = fcomposites less than 30g.
a List sets R and S.
b Find R \ S and R [ S.
c Represent the sets R and S on a Venn diagram.
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the elements of set:
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b B
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c A0
f A[B
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108
SETS AND VENN DIAGRAMS
(Chapter 3)
7
A
h
j
b
g
c
d
a
i
This Venn diagram consists of three overlapping circles
A, B and C.
a List the letters in set:
i A
ii B
iii C
iv A \ B
v A[B
vi B \ C
vii A \ B \ C
viii A [ B [ C
B
k
fe
l
C
U
b Find:
i n(A [ B [ C)
ii n(A) + n(B) + n(C) ¡ n(A \ B) ¡ n(A \ C) ¡ n(B \ C) + n(A \ B \ C)
F
VENN DIAGRAM REGIONS
We use shading to show various sets being considered.
For example, for two intersecting sets A and B:
A
A
B
U
A
B
U
U
B
U
0
A \ B is shaded
A is shaded
A
B
A \ B 0 is shaded
B is shaded
Example 11
Self Tutor
On separate Venn diagrams, shade these regions for two intersecting sets A and B:
a A[B
b A0 \ B
c (A \ B)0
a
b
c
A
A
B
U
A
B
U
(in A, B or both)
B
U
(outside A, intersected
with B)
(outside A \ B)
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Click on the icon to practise shading regions representing various subsets. You
can practise with both two and three intersecting sets.
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DEMO
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SETS AND VENN DIAGRAMS
(Chapter 3)
109
EXERCISE 3F
1
A
Draw separate Venn diagrams showing shading regions for:
a A\B
b A \ B0
PRINTABLE
0
c A [B
d A [ B0
VENN DIAGRAMS
(OVERLAPPING)
e (A \ B)0
f (A [ B)0
B
U
2
A
Suppose A and B are two disjoint sets. Shade on separate
Venn diagrams:
a A
b B
PRINTABLE VENN
0
0
DIAGRAMS (DISJOINT)
c A
d B
e A\B
f A[B
0
g A \B
h A [ B0
0
i (A \ B)
B
U
Suppose B µ A, as shown in the given Venn diagram.
Shade on separate Venn diagrams:
PRINTABLE VENN
a A
b B
DIAGRAMS (SUBSET)
0
0
d B
c A
e A\B
f A[B
h A [ B0
g A0 \ B
i (A \ B)0
3
A
B
U
4
This Venn diagram consists of three intersecting sets. On
separate Venn diagrams shade:
PRINTABLE VENN
a A
b B0
DIAGRAMS (3 SETS)
c B\C
d A[B
e A\B\C
f A[B[C
0
h (A [ B) \ C
g (A \ B \ C)
i (B \ C) [ A
B
A
C
U
G
NUMBERS IN REGIONS
We have seen that there are four regions on a
Venn diagram which contains two overlapping
sets A and B.
A
B
A \ B 0 A \ B A0 \ B
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A0 \ B 0
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110
SETS AND VENN DIAGRAMS
(Chapter 3)
There are many situations where we are only interested in the number of elements of U that
are in each region.
We do not need to show all the elements on
the diagram, so instead we simply write the
number of elements in each region in brackets.
For example, the Venn diagram opposite
shows there are 4 elements in both sets A and
B, and 3 elements in neither set A nor B.
Every element in U belongs in only one region
of the Venn diagram. So, in total there are
7¡+¡4¡+¡6¡+¡3¡=¡20 elements in the universal
set U.
A
B
(7)
(4)
(6)
(3)
U
Example 12
Self Tutor
P
If (3) means that there are 3 elements in the set
P \ Q, how many elements are there in:
Q
(7)
a P
c P [Q
e Q, but not P
(3) (11)
(4)
U
b Q0
d P , but not Q
f neither P nor Q?
a n(P ) = 7 + 3 = 10
b n(Q0 ) = 7 + 4 = 11
c n(P [ Q) = 7 + 3 + 11
= 21
d n(P , but not Q) = 7
e n(Q, but not P ) = 11
f n(neither P nor Q) = 4
Venn diagrams allow us to easily visualise identities such as
n(A \ B 0 ) = n(A) ¡ n(A \ B)
A
0
n(A \ B) = n(B) ¡ n(A \ B)
B
A \ B0
U
EXERCISE 3G
If (2) means that there are 2 elements in the set
A \ B, give the number of elements in:
1
B
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a B
c A[B
e B, but not A
(5)
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(7)
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b A0
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f neither A nor B
IB_STSL-2ed
SETS AND VENN DIAGRAMS
2
X
(6)
(3)
(2)
U
In the Venn diagram given, (a) means that there
are a elements in the shaded region.
Notice that n(A) = a + b.
Find:
a n(B)
b n(A0 )
c n(A \ B)
0
d n(A [ B) e n((A \ B) ) f n((A[ B)0 )
3
A
B
(a)
(b)
(c)
(d)
U
4
P
The Venn diagram shows that n(P \ Q) = a and
n(P ) = 3a.
Q
(2a) (a) (a+4)
a Find:
i n(Q)
iii n(Q0 )
(a-5)
U
111
Give the number of elements in:
a X0
b X \Y
c X [Y
d X, but not Y
e Y , but not X
f neither X nor Y
Y
(8)
(Chapter 3)
ii n(P [ Q)
iv n(U )
b Find a if:
i n(U ) = 29
ii n(U ) = 31
Comment on your results.
Example 13
Self Tutor
Given n(U ) = 30, n(A) = 14, n(B) = 17 and n(A \ B) = 6, find:
a n(A [ B)
b n(A, but not B)
A
We see that b = 6
a + b = 14
b + c = 17
a + b + c + d = 30
B
(a)
(b)
(c)
(d)
U
fas
fas
fas
fas
n(A \ B) = 6g
n(A) = 14g
n(B) = 17g
n(U) = 30g
Since b = 6, a = 8 and c = 11.
So, 8 + 6 + 11 + d = 30 and hence d = 5.
a n(A [ B) = a + b + c = 25
b n(A, but not B) = a = 8
5 Given n(U ) = 26, n(A) = 11, n(B) = 12 and n(A \ B) = 8, find:
a n(A [ B)
b n(B, but not A)
6 Given n(U ) = 32, n(M) = 13, n(M \ N ) = 5 and n(M [ N ) = 26, find:
b n((M [ N)0 )
a n(N)
7 Given n(U ) = 50, n(S) = 30, n(R) = 25 and n(R [ S) = 48, find:
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112
SETS AND VENN DIAGRAMS
(Chapter 3)
H
PROBLEM SOLVING
WITH VENN DIAGRAMS
In this section we use Venn diagrams to illustrate real world situations. We can solve problems
by considering the number of elements in each region.
Example 14
Self Tutor
A squash club has 27 members. 19 have black hair,
14 have brown eyes, and 11 have both black hair and
brown eyes.
a Place this information on a Venn diagram.
b Hence find the number of members with:
i black hair or brown eyes
ii black hair, but not brown eyes.
a Let Bl represent the black hair set and Br represent the brown eyes set.
a + b + c + d = 27
Bl
Br
a + b = 19
b + c = 14
(a) (b) (c)
b = 11
(d_)
) a = 8, c = 3, d = 5
U
Bl
(8)
i n(Bl [ Br) = 8 + 11 + 3
= 22
b
Br
(11) (3)
ii n(Bl \ Br0 ) = 8
(5)
U
Example 15
Self Tutor
A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive
from 5 m. How many dive from both platforms?
Let T represent those who dive from 10 m and
F represent those who dive from 5 m.
d=0
F
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b + c = 17
a + b + c = 25
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100
(b)
75
(a)
75
T
fas all divers in the squad must dive
from at least one of the platformsg
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SETS AND VENN DIAGRAMS
113
n(both T and F ) = n(T \ F )
= 10
F
T
(Chapter 3)
(8) (10) (7)
So, 10 members dive from both platforms.
(0)
U
EXERCISE 3H
1 Pelé has 14 cavies as pets. Five have long hair
and 8 are brown. Two are both brown and have
long hair.
a Place this information on a Venn diagram.
b Hence find the number of cavies that:
i do not have long hair
ii have long hair and are not brown
iii are neither long-haired nor brown.
During a 2 week period, Murielle took her umbrella
with her on 8 days. It rained on 9 days, and Murielle
took her umbrella on five of the days when it rained.
a Display this information on a Venn diagram.
b Hence find the number of days that:
i Murielle did not take her umbrella and it
rained
ii Murielle did not take her umbrella and it
did not rain.
2
3 A badminton club has 31 playing members. 28 play singles and 16 play doubles. How
many play both singles and doubles?
4 In a factory, 56 people work on the assembly line. 47 work day shifts and 29 work night
shifts. How many work both day shifts and night shifts?
5
A
Use the figure given to show that:
n(A [ B) = n(A) + n(B) ¡ n(A \ B)
B
(a)
(b)
(c)
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114
SETS AND VENN DIAGRAMS (Chapter 3)
Example 16
Self Tutor
A city has three football teams A, B and C, in the national league.
In the last season, 20% of the city’s population went to see team A play, 24% saw team
B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C,
and 6% saw both B and C. 1% saw all three teams play.
Using a Venn diagram, find the percentage of the city’s population which:
a saw only team A play
b saw team A or team B play but not team C
c did not see any of the teams play.
We construct the Venn diagram in terms of percentages.
Using the given information,
A
a=1
a+d =4
a+b=6
a+c =5
B
(d)
(g)
(c)
(a)
(e)
(b)
(f)
(h)
saw
saw
saw
saw
all three teams playg
A and Bg
B and Cg
A and Cg
) d = 3, b = 5 and c = 4
C
U
f1%
f4%
f6%
f5%
Also, 20% saw team A play,
so g + 1 + 4 + 3 = 20 ) g = 12
A
B
(3)
(g)
(4)
(1)
(e)
(5)
(f)
(h)
) e = 15
28% saw team C play,
so f + 1 + 5 + 4 = 28
) f = 18
In total we cover 100% of the population,
so h = 42.
C
U
24% saw team B play,
so e + 1 + 5 + 3 = 24
A
a n(saw A only) = 12% fshadedg
B
(3)
(12)
(4)
(1)
(15)
b
(5)
(18)
(42)
n(A or B, but not C)
= 12% + 3% + 15%
= 30%
c n(saw none of the teams) = 42%
C
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d Physics but not Chemistry?
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6 In a year group of 63 students 22 study Biology, 26 study Chemistry, and 25 study
Physics. 18 study both Physics and Chemistry, four study both Biology and Chemistry,
and three study both Physics and Biology. One studies all three subjects. How many
students study:
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SETS AND VENN DIAGRAMS (Chapter 3)
115
7 36 students participated in the mid-year adventure trip. 19 students went paragliding,
21 went abseiling, and 16 went white water rafting. 7 did abseiling and rafting, 8 did
paragliding and rafting, and 11 did paragliding and abseiling. 5 students did all three
activities. Find the number of students who:
a went paragliding or abseiling
b only went white water rafting
c did not participate in any of the activities mentioned
d did exactly two of the activities mentioned.
8 There are 32 students in the woodwind section of the school orchestra. 11 students can
play the flute, 15 can play the clarinet, and 12 can play the saxophone. Two can play
the flute and the saxophone, two can play the flute and the clarinet, and 6 can play the
clarinet and the saxophone. One student can play all three instruments. Find the number
of students who can play:
a none of the instruments mentioned
b only the saxophone
c the saxophone and the clarinet but not the flute
d only one of the clarinet, saxophone, or flute.
9 In a particular region, most farms have livestock and crops. A survey of 21 farms showed
that 15 grow crops, 9 have cattle, and 11 have sheep. Four have sheep and cattle, 7 have
cattle and crops, and 8 have sheep and crops. Three have cattle, sheep, and crops. Find
the number of farms with:
a only crops
b only animals
c exactly one type of animal, and crops.
REVIEW SET 3A
1 If S = fx j 2 < x 6 7, x 2 Z g:
a list the elements of S
b find n(S).
2 For each of the following numbers, state if they are rational or irrational. If the
number is rational, prove your claim.
a
4
7
b 0:165
p
e
5
d 56
c
8
0
f 0:181 818 ::::
3 Determine whether A µ B for the following sets:
a A = f2, 4, 6, 8g and B = fx j 0 < x < 10, x 2 Z g
b A = ? and B = fx j 2 < x < 3, x 2 R g
c A = fx j 2 < x 6 4, x 2 Q g and B = fx j 0 6 x < 4, x 2 R g
d A = fx j x < 3, x 2 R g and B = fy j y 6 4, y 2 R g
4 Find the complement of X given that:
a U = fthe 7 colours of the rainbowg and X = fred, indigo, violetg
b U = fx j ¡5 6 x 6 5, x 2 Z g and X = f¡4, ¡1, 3, 4g
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116
SETS AND VENN DIAGRAMS (Chapter 3)
5 On separate Venn diagrams like the one alongside,
shade:
a N0
b M \N
c M \ N0
M
N
U
6 Let U = fthe letters in the English alphabetg, A = fthe letters in “springbok”g,
and B = fthe letters in “waterbuck”g.
a Find:
i A[B
ii A \ B
iii A \ B 0
b Write a description for each of the sets in a.
c Show U , A and B on a Venn diagram.
7 Let U = fx j x 6 30, x 2 Z + g, P = ffactors of 24g, and Q = ffactors of 30g.
a List the elements of:
i P
ii Q
iii P \ Q
b Illustrate the sets P and Q on a Venn diagram.
iv P [ Q
8 A school has 564 students. 229 of them were absent for at least one day during
Term 1 due to sickness, and 111 students missed some school because of family
holidays. 296 students attended every day of Term 1.
a Display this information on a Venn diagram.
b Find how many students:
i missed school for both illness and holidays
ii were away for holidays but not sickness
iii were absent during Term 1 for any reason.
9 The main courses at a restaurant all contain rice or onion. Of the 23 choices, 17
contain onion and 14 have rice. If Elke dislikes onion, how many choices does she
have?
10 38 students were asked what life skills they
had. 15 could swim, 12 could drive and 23
could cook. 9 could cook and swim, 5 knew
how to swim and drive, and 6 could drive
and cook. There was a student who could
do all three. Find the number of students
who:
a could cook only
b could not do any of these things
c had exactly two life skills.
11 Consider the sets U = fx j x 6 10; x 2 Z + g, P = fodd numbers less than 10g,
and Q = feven numbers less than 11g.
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a List the sets P and Q:
b What can be said about sets P and Q?
c Illustrate sets P and Q on a Venn diagram.
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SETS AND VENN DIAGRAMS (Chapter 3)
117
REVIEW SET 3B
1 True or false:
a N ½Q
b 02Z+
e Z + \ Z ¡ = f0g
d R µQ
2
c 02Q
a Write in set builder notation:
i the real numbers between 5 and 12
ii the integers between ¡4 and 7, including ¡4
iii the natural numbers greater than 45.
b Which sets in a are finite and which are infinite?
3 List the subsets of f1, 3, 5g.
4 Let U = fx j 0 < x < 10, x 2 Z g, A = fthe even integers between 0 and 9g,
and B = fthe factors of 8g.
a List the elements of:
ii A \ B
i A
iii (A [ B)0
b Represent this information on a Venn diagram.
5 If S and T are disjoint sets, n(S) = s and n(T ) = t, find:
a S\T
b n(S [ T )
6
Give an expression for the region shaded in:
A
B
a blue
b red.
C
U
7 In a car club, 13 members drive a manual and 15 members have a sunroof on their
car. 5 have manual cars with a sunroof, and 4 have neither.
a Display this information on a Venn diagram.
b How many members:
i are in the club
ii drive a manual car without a sunroof
iii do not drive a manual?
8 All attendees of a camp left something at home. 11 forgot to bring their towel, and
23 forgot their hat. Of the 30 campers, how many had neither a hat nor a towel?
9 Consider the sets U = fx j x 6 60, x 2 Z + g, A = ffactors of 60g,
and B = ffactors of 30g.
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c Illustrate sets A and B on a Venn diagram.
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118
SETS AND VENN DIAGRAMS (Chapter 3)
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10 At a conference, the 58 delegates speak
many different languages. 28 speak
Arabic, 27 speak Chinese, and 39 speak
English. 12 speak Arabic and Chinese,
16 speak both Chinese and English, and
17 speak Arabic and English. 2 speak
all three languages. How many delegates
speak:
a Chinese only
b none of these languages
c neither Arabic nor Chinese?
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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_03\118IB_STSL-2_03.CDR Thursday, 11 February 2010 10:15:05 AM PETER
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