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Unformatted text preview: THE LANGUA Sets A set is any collection of objects. A set may be defined by listing the members:
{1.2.3.4}. {1,3,5,7,9,...}. or by describing them: {square numbers}. {1: —l <: x -..<.. 4}:
The universal set is the set of all the objects under
discussion. and the null set is the set which contains ' no members.
lfall the members ofa set A belong to a set 3. then AisasuhsetofB. The following symbols are used. {...} thesetofall... {xz . . the set of all x such that . ..
E the universal set
¢ thc_null set xEA xisamemberolA
A c: B AisasubsetolB Sets may be illustrated in Venn diagrams. as in
Fig l. 1—1 Union chxeL Intersection of sets Problem
The Headmistress of loinspro College called to her office the ll girls in the hockey team and the
7 girls in the netball team. There were 3 girls who
were in both teams. How many girls came to the
Headmistress's office? A Venn diagram can help to make the situation clear. Let
E: {Students atJoinspro College}.H = {students
in the hockey team} and N _= {students in the
netball team}. Here is the set H: 'E. GE l: SETS Mme Here is the set N: Fig 3 And here is the set of students who came to the
Headmistress‘s office: From Fig‘l we can see that the total number of
students who came to the Headmistress‘s office was 15. This set. which is made up of all the students in
the hockey team together with all those in the net-
ball team. is called the union ofthe sets H and N. We
might say that the two sets have been united into
one set. This set is written H U N. The union of two sets is the set formed by putting
the two sets together. The union ofA and B is written
A u B. New look again at Fig 4 If Mary was one of the
girls who came to see the Headmistress. can you tell
which team she belonged to? There are three
possibilities: either
1. Mary belonged to H (and not to N); or
2. Mary belonged to N (and not to H); or
3. Mary belonged to both H and N. In other words. Mary belonged either to H or to
N or to both. This is true of all the members of HUN. This gives us another way of Iboking at the union
of two sets: A U B = {objects which belong to A or B(or both)}. .-+.. a.-__"l-—I--H—H - This set is 3110“ by {ht Shaded “aim in Fig 5 a) List the members of the union ofA =} {1. 3.
4. 5. 7} and B = {3, 4, 8. 9}.
Draw a Venn diagram showing the sets A. B
and A u B. ( c) If two sets are disjiSlnt. now can number of members in their union?
ll two sets overlap. what can you say about
the number of members in their union? “‘5 A U a :-. __ __ ._ - I
8 Let = fir“ Joinspro
- - - - - College}
I t Z '
Compare this With the meaning of in ersec ton F = {those who Study French}!
The intersecfim or two Sets 15 G = {thosc who study Gerography}'I
set of all objects which belong to both sets. Th1. and H = {those who study History}
set in ehown by the broken line in Describe the members oftiie following sets.
Fig 4 iit contains the 3 students who were In (a) F l»! (b) F r‘LG.
both teams. I
The intersection of A and B is
written 'A n B . So :
A n B = {objects which belong to both A and B}. This set is shown by the shaded region in Fig 6 . 'l-l @Let MK: {multiples of 7:; For example.
Mg}: {6.l2.18.24.30....}. (it) Find the first five membersof: (b) Find the valuesofx andzin the Following
" equations. _ Main M74.- Mx
(unit/{61:1 M = M Z
@a) Let:
T = {t. h. e}. A = {a. s. o. c. i. t. V. c}. ' a... h__-.___..._.._ .l. __H_ P = -{p..r.0.c.t.y}.
@Write down the members ot the Intersections of Find:
the following pairs ofsets. (i) T n A_ (ii) A n p,
(a) {1.4.9.16.25.36}.{2.4.3.i5i32~54i (iii) (Tn A) n p, (iv) Tn (A n p)_
(b) {a &13_.£;.17*"+}_{a, e‘ i. o, u} What do you notice about (iii) and (iv)? (d) {whole numbers n : n 2.. 6},
{whole numbers n : n s. 9} . _ (c) If a,b and e are scientific?! of - * -. a set S, and " an operation
- @..'Two sets A and'Bmure dfsyomr it"A n B = or'E’ii- “mh that
press this relation in words; . (a I b) u c I. l e (b e c)
Which oftite followingpairsofsetsaredisjoint? is true-Jot all member: of S,
(:1) {states in Africa}. we say that the operation '
{Nigerim Indonesia. Canada. France} is assoc iative. (b) {students in your class}. {students in Form 5} (C) {prime numbers}. {even numbers} (d) {reCtangles}. {squares} _ _ (4') iPOims OUISidca Cumin: (ii) In union of sets assoc-
{points inside the cube} - iative ? i‘ (1) II intersection of set!
associative 1' In Fig 8. E. = {students in your school} and
A = {students in your class}. What can you say
‘.bout the students represented by points in the -- region inside I. but outside A? The region inside 8 and outside A represents all
the objects under discussion which are not ID A. t Fig 3 This set is called the complement of A. writtenGt
The set Qt is shaded horizontally in Fig 9, If A is a subset of a universal set 1, the complement
of A is the set of objects which belong to E but do not belong to A;" .
Here is another example. If E. = {whole numbers},
the complement of {even numbers} is {numbds which are not even}. The complement is therefore
{ odd numbers}. I; @Let {months}.
A = {those whose names contain the
letter ‘a'}. .
B = {those whose names end in ‘ber‘}.
and T = {those with 31 days}.
(a) List the members ofC‘l ,$ and CT . (b) Describe the members oIQ andfT . II a = {rational numbers}, write down the complements of the following sets.
(a){x:x< l} (b){y:y$-—3}
(c) {2: z is greater than 4} (b) If E = {students at Jornspro‘C'ElIe'geY. de- scribe the members of the complements of
the following sets. B = {those who have two or more
brothers} C = {those who are over 15 years old} 1—3 Classification At the beginning of this chapter. we represented
the hockey and netball teams of Joinspro College
on a Venn diagram. In Fig l0. E= {students at
Joinspro College}. H = {members of the hockey
team} and N = {members of the netball team}.
The two subsets H and N divide the universal set
linto four regions. which we have called I to IV. E /f M Fig 10 Region I is inside both H and N. It is H n N. the
intersection of H and N. It represents {students who
are in the hockey team and the netball team}. Now look at region II. All the points of this set
are inside H but outside N. It represents {students
who are in the hockey team but not in the netball
team}. In Fig 11_H is shaded horizontally and'gtl is
shaded vertically. You can see that region II is the
intersection of these two sets. that is. H n i ' Fig 11 What can you say about the students in rEgions
III and IV in Fig lO‘IDescribe these regions in‘terms
of H and N. In this example, there are two properties under 3 .
consideration: being in the hockey team. and being - in the netball team. A student may have two: one or none or these properties. The two properties
therefore divide the universal set into four defies: Class I
In the hockey team and in the netball team Inside H and inside N (H n N) Class H
I_r_t the hockey team but not in the netball team Inside H and outside N (H rug-V) Class III
Not in the hockey team but in the netball team Outside H and inside N qs F N’) Class IV We say that we have classified the students in
Joinspro College according to whether or'lbt they
are in the hockey or netball teams. Classification is
a very common human activity; it is especially
important to librarians and biologists. even numbers}
numbers divisible by 3}. ll
—-*-—..-o—-t (a) List the members of A = {even numbers
which are divisible by 3 }. Put the members of A intothe correct region of your diagram (b) List the members of B = {odd numbers
which are divisible by 3}. Put these members i into your Venn diamm. tc) List the members of C = {odd numbers
which are not divisiblelby 3}. Put these
membersjnto your diagram. '(d) Write the members of the remaining region
on "your diagram. How can you describe this
set? @111 Fig [4, E. = {names of months}.
J =_: {those which begin with J},
and Y it {those which end with y}. 3' Y (b) List the members of these classes. (c) Express each class in terms ofJ and Y. —I-l 1—4 Th res-set problems Copy Fig 15. in which
‘8. = {students at Abrakabra School},
A = {those who study Art}.
3 = {those who study Biology}, and C = {those who study Chemistry}.
Shadetheseu A n = {those who study Art but not Biology} Hence find the set (A m B) n C. What subjects do: the members of this set study?
Copy Fig 15.again. but this time shade the set: [B n C = {those who study Chemistry but not
Biology}. Hence find the set A n (E n C). What subjects do
the members O‘fthis set study? Fig 15 students at Abrakabra School. There are now eight
classes, given by the eight regions of’the Venn
diagram. We have labelled these 1 to VIII in Fig 16 For example. class III is the set A Fifi n C. This
region is inside A. outside 3 and inside 0' the i students in this class therefore study Art and
Chemistry but not Biology. Questions (Q‘- 1‘) 1. Which class ith n 3 mg: 2’ What subjects do
the students in this class study? 2. What subjects do the students in class VIII
study? Express this class in terms ol'A, B and C. 3. hich classes orsmdcms study Biology? Which
classes study Chemistry? Which classes study
Biology and Chemistry? @111 Fig 17,1ct
Z
. F 0:
andS— Fig 17 -— {those who can swim} {students at Joinspro College}.
{girls},
{those over 16 years old}. (3) Which regions represent the following
classes of student? A -—~ {girls over 16 who cannot swim} 3 {boys over 16 who can swim} C = {girls aged 16 or under who cannot
swim} (b) List the members oftne set J nU’ n F and
the members ol'the setp' n Y ref. Fa. —|. Number of members in a set In many classification problems. we want to find
the number of members in various classu. We refer to the number of members inlthe set A as
rt(A). In Fig 13.for example. MA n? n C) = 5‘ Question
In Fig 18, what are (a) [email protected] n‘ B (b) n(A m C), (c) n(A u C)? (. 5 )A ncwspaper boy supplies three different pap-en X. Y and 2. He sells allhe has to 200 customers.
with the following results. (a) No one bought more than one copy of the
same paper. (b) Of those who bought only onepaper, 44
bought X and 36 bought Y. “ (C) or “1053 who bought two papers only, #9 bought X and Y, 12 bought X and Z and
16 bought Y and Z. (d) Six people bought all three papers. Calculate how many copies of each paper were .
sold, and how many customers bought 2 only. ...
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- Spring '09
- Johnson
- Algebra
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