Download + 1 sO - Department of Mathematics, CCNY

6
SETS / Chap. 1
since neither set contains any elements. Therefore we are justified in
referring to the empty set or the null set. * We denote the null set by
the special symbol 0.
1.3. To illustrate the inadequacy of displaying a few elements of a set and
indicating the other elements by three dots, consider the set A of all
numbers of the form
n2
PROBLEMS
1.1. We list eight sets. For each set, state whether it is finite or infinite.
If finite, count the number of elements in the set. Where feasible, write
the set using the roster method.
(a) The set of footnotes in Section 1.
(b) The set of letters in the word "probability."
(c) The set of odd positive integers.
(d) The set of prime numbers less than one million.
(e) The set of paths by which the following figure can be traced without
lifting one's pencil or retracing any line segment:
3
1
2
4
Figu.re 3
(f) The set of those points (in a given plane) that are exactly five units
from the origin O.
(g) The set of real numbers satisfying the equation X2 - 3x + 2 = O.
(h) The set of possible outcomes of the experiment in which one card
is selected from a standard deck of 52 cards.
1,2. The following paragraph was written by a student impressed with the
technical vocabulary of set theory. Rewrite in more usual English prose.
Let C be the set of Mr. and Mrs. Smith's children. C was equal to 0
until March 1, 1958. C contained exactly one element from that date
until March 15, 1959 when it increased its membership by two!
,..The following true story concerns the attempt of a well-known professor of
mathematics to teach his five-year-old son the subtle distinction between "a"
(or "an") and "the." One day the son answered the telephone, listened a moment
and then said, "I'm sorry, but you have the wrong number." (Isn't this what
most of us say when someone dials incorrectly?) The father, having overheard,
immediately called the boy to him and gently instructed, "What you said would
be correct if there were exactly one wrong uumber. But since there are many
possible wrong numbers, it would be more accurate to say, 'I'm sorry, but you
have a wrong number.' "
7
Sec. 1 / EXAMPLES OF SETSi BASIC NOTATION
+ (n -
l)(n
- 2)(n - 3),
where n is any positive integer. Show that the first three elements
(i.e., those obtained when n = 1, 2, 3) are 1, 4, and 9, so that one is
tempted to write A = {I, 4, 9, ... }. If A is written this way on an 1. Q.
test, we do not hesitate to write the next element as 16. But show that
the next element (obtained when n = 4) is actually 22 and not 16!
Indeed, it is possible to write a defining property for a set so that its
fourth element (in order of magnitude) is any number, say 94, although
its first three elements are 1, 4, 9. Formulate such a defining property.
1.4. Let A
=
{O, 1,2,3, 4}. List the elements, if any, of each of the follow-
ing sets:
(a) {x
(c) {x
(e) {x
(g) {x
A I 2x - 4 = O}
A I XS - 4X2 + 3x = O}
E A I z + 1 > O}
E A I X2 - 5x + 4 ~ O}
E
E
(b) {x E A I X2
(d) {z e A I X2
(f) {x E AI2x
(h) {x E A I X2
4 = O}
O}
+ 1 O}
- X < O}
-
=
s
1.6. Let x and y be the coordinates of a point in the plane. Identify the
following sets and give a geometric interpretation of your results:
(a) {(x, y) I x
(b) {(x, y) I x
(c) {(x, y) I x
+ y = 5 and
+ y = 5 and
+ y = 5 and
3x - y = 3}
2x + 2y = 3}
2x + 2y = 1O}
.;1;.6. Show that set equality has the following properties:
(i) Set equality is a reflexive relation; i.e., A = A for any set A.
(ii) Set equality is a symmetric relation; i.e., for any sets A and B, if
A = B, then B = A.
(ill) Set equality is a transitive relation; i.e., for any sets A, B, and
if A = Band B = C, then A = C.
a,
(Note: A relation that is reflexive, symmetric, and transitive is called
an equivalence relation.)
1.7. Determine whether A
= B or A ;e B.
(a) A = {2, 4, 6}, B = {4, 6, 2}.
(b) A = {I, 2, 3}, B = {Mars, Venus, Jupiter}.
(c) A = {* I * is a plane equilateral triangle}, B = {* I * is a plane
equiangular triangle}.
(d) A = {x I X2 - 2x + 1 = O}, B = {z I x - I = OJ.
(e) A = {x 12x2 - 5x
2 = OJ, B = {x 12x3 - 5x2 2x = OJ.
+
+
f
"i
12
SETS / Chap. 1
than a million different subsets. Ordinarily, however, as the following examples point out, one is interested in studying a small number
of subsets from among the many available ones.
Example
the set of
merate as
die showed
2.4. Let a green and a red die be rolled, and let S denote
possible outcomes. S has 36 elements which we can enufollows, using the abbreviation
(x, y) to stand for "green
the number x and red die showed the number y":
(1, 1)
(2, 1)
(6,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
By Theorem 2.1, there are 236 subsets of S. But relatively few of
these subsets have any special interest, even to players of "craps."
Some of these are:
(i) SI = the subset made up of those outcomes for which the sum
of the numbers on the two dice is 7; i.e.,
SI=
{(I, 6), (2,5), (3,4), (4,3), (5,2), (6, I)}.
(ii) S2 = the subset containing the outcomes for which the sum of
the numbers on the two dice is 11; i.e.,
82 == {(5, 6), (6,5)}.
(iii) 83 = the subset containing the outcomes for which the sum
of the numbers on the two dice is either 7 or 11; i.e.,
13
See, 2 / SUBSETS
subset of those entries containing the names of women who are residents of her dormitory, the mathematics
department must estimate
ahead of time the approximate number in the subset of entries naming
students who will elect mathematics
courses, a student may be especially interested in the subset of entries naming aU freshmen who
corne from his hometown, etc. If the information for each student is
entered by punching holes on certain specially-designed
cards, then
the cards corresponding to these subsets of A, as well as many others,
can be sorted out of the whole set of cards by a machine. In fact,
such sorting machines are designed for the Pill' pose of speedily select. ing certain subsets from a given set.
We conclude with an example designed to test the reader's grasp
of the difference between the notions of set membership (symbolized
bye) and set inclusion (symbolized by C).
Example 2.6. Consider the set M of majorities in a committee of
four individuals, each having one vote. Let us label the individuals
a, b, c, d, and note that a majority is itself a set of three or more of
these committeemen.
Thus, the set M has sets as elements, and we
write 111 using braces within braces:
M
=
{{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}}.
Thus {a, b, c} is an element of M, but although a set, it is not a subset
of M. (Why?) But {{a, b, c}} is a subset of M since its only element,
{c, b, c}, is indeed also an element of M.
PROBLEMS
S3 = {(I, 6), (2,5), (3,4), (4,3), (5,2), (6, 1), (5,6), (6, 5)}.
Whenever any experiment is performed (as in this example), we can
think of the set of all possible outcomes of the experiment.
We shall
see that such sets and their subsets are of great importance in the
mathematical
theory of probability.
Example 2.5. The annual directory of college X lists the name,
hometown, college residence, and telephone number of each of the
college's 2000 students. Let A be the set of these 2000 entries, each
entry containing the four pieces of information described above. The
total number of subsets of A is astronomical,
being 22000• But the
housemother in a certain dormitory is mainly concerned with the
2.1. Let S be the set of 36 outcomes of the experiment in which a green
and a red die are rolled. (See Example 2.4.) We define certain subsets
of S by listing their elements. State a defining property for each subset.
(a) {(I, 1), (2,2), (3,3), (4,4), (5,5), (6, 6)}
(b) {(I, 1), (1,2), (1,3), (1,4), (1,5), (1, 6)}
(c) {(I, 3), (2, 2), (3, I)}
(d) {(I, 3), (1, 4), (1,5),(1,6), (2, 4), (2,5), (2,6), (3, 5), (3,6), (4, 6)}
2.2. Let A = {I, 2, 3}. Identify the sets B such that {I} C B, B ~ A, and
B r! A.
2.3. You are told that there is only one set A such that A C B. Identify
the set B.
14
SETS I Chap. 1
2.4. Let A be any set. The set {X! X C A} of all subsets of A is called
the power set of A and is denoted by 2A. (If A is a set with n elements,
then Theorem 2.1 says that the power set 2A has 2n elements. This fact
accounts for the name "power set" and the symbol 2A used to denote this set.) Explain the following true statements, assuming
A = {x, y, z}:
(a)
(b)
(c)
(d)
(e)
0 t A, but 0 f 2A.
x
E 2A.
A is an element, but is not a subset of 2A.
{A} is not an element, but is a subset of 2A.
2.6.. Which of the following are correct and why?
E
E
{{I}}
{I, {I}}
(b) {I} C {{I}}
(d) {I} ~ {I, {I}}
2.6. Give an example of two sets A and B such that both A
A C B are true.
€
Band
2.7. The graph of the set of points C = {(x, y) I XZ + yZ = 4}, where x and
yare real numbers, is the circumference of the circle with center at
(0, 0) and radius 2 units. Determine the graphs of the following subsets of C:
(a) {(x, y)
(c) {(x, y)
(e) {(x, y)
E
C Ix
E
C
E
C
! :1'
!y
=
O}
= 3}
~ O}
15
telephone in that exchange. The telephone dial contains ten slots, one
for each of the digits 0, 1, 2, "', 9, but letters appear in only eight
of these slots. If the first number cannot be a zero, how many different
telephone numbers, distinguishable as dialed, are possible?
2.14. How many three-digit even integers can be formed from the digits
1, 5, 6, and 8, with no digit repeated?
2.16. How many different ways are there of selecting two letters from the
set {a, b, c}? Let the reader realize that the question as siaied ie vague
and needs to be made precise before it can be answered. We must know
how the letters are selected, and we must decide when results of the
selection process will be considered different. We list four possibilities,
Answer the question in each case.
A, but X 1'2A.
{x, y} f A~ but {r, y}
E
(a) {I}
(c) {I}
Sec. 2 I SUBSETS
(b) {(x, y) € C ! x = 2}
(d) {(x, y) € C ! x > O}
(f) {(x,y)€C!y=V4-xZ}
2.8. (a) If Z E A and A ~ B, is it necessarily the case that z € B?
(b) If z € A and A € B, is it necessarily the case that Z E B?
2.9. Draw a tree diagram to illustrate the fundamental principle of counting,
assuming n = 3 and NI = 4, Nz = 3, N« = 2.
2.10. Assume the truth of the fundamental principle of counting for n = 2,
i.e., for a job made up of only two tasks. Prove the principle for any
positive integer n by mathematical induction.
In each of the following problems, state explicitly how the fundamental principle of counting is used in obtaining your answer. Draw a tree diagram where
feasible.
2.11. A man has five coins in his pocket. He agrees to give one coin to his
son and one to his daughter. In how many ways can this be done?
2.12. In how many different orders can one call out the numbers 1, 2, 3, 4, 5?
2.13. In dialing a telephone number, one has to select seven slots, the first
two fer the letters of the exchange, and then five digits to identify the
(a) The first letter is chosen and the second is selected from the remaining two letters; i.e., repetitions are not allowed. We count two
ways of making the selections different if they result in different
ordered pairs of letters; i.e., we record not only which two letters
were selected, but also the order in which they were selected.
(b) The first letter is chosen and the second is selected from the entire
set of three letters; i.e., repetitions are allowed. We count ordered
pairs of letters as in (a).
(c) Repetitions are not allowed, as in (a), and we count two ways of
making the selections different only if they result in different sets
of two letters; i.e., we disregard the order in which the letters were
selected.
(d) Repetitions are allowed, as in (b), and we disregard order I1S ill (c).
2.16. How many different ways are there of selecting two cards from a
standard deck of 52 cards? Consider various interpretations of this
question, as in the preceding problem.
2.17. How many ways can three coins fall? four coins? n coins, where n
is any positive integer?
2.18. Two cards are drawn one after the other from a standard deck of 52
card.s. In how many ways can one draw
(a) first a spade and then a heart?
(b) first a spade and then a heart or a diamond?
(c) first a spade and then another spade?
2.19. Repeat the preceding problem, assuming the first card is put back in
the deck before the second is drawn.
2.20. Let A = {I, 2,3, .. " 365}. (a) Two numbers are selected in order,
each from the full set A. The result is an ordered 2-tuple, or ordered
pair of numbers. How many are there? (b) Three numbers are selected
SETS ! Chap. 1
16
in order, each from the full set A. The result is an ordered S-tuple,
or ordered triple of numbers. How many are there? (c) r numbers are
selected in order, each from the full set A, where r is some positive
integer. The result is an ordered r-tuple. How many are there? (Note:
A general definition of an ordered r-tuple is given in Section 5.)
2.21. (a) How many ways are there of placing three distinguishable balls
into two numbered cells? Into three numbered cells? Into n numbered eells?
(b) How many ways are there of placing r distinguishable balls into
two numbered cells? Into three numbered cells? Into n numbered
eells?
3. Operations on sets
In any particular discussion of sets, it is necessary to define some
fixed set of elements (called the universal set) to which we limit the
discussion. This point has been eloquently made by Langer:
In ordinary conversation, we assume the limitations of such a universe,
as when we say: "Everybody knows that another war is coming," and.
assume that "everybody" will be properly understood to refer only to
adults of normal intelligence and European culture, not to babies intheir
cribs, or the inhabitants of remote wildernesses. For conversational purposes, the tacit understanding will do; but if the statement is to be challenged, i.e., if someone volunteers to produce a person to whom it is not
true, then it becomes important to know just what the limits of its applicability really are. Arguments of this sort have their own technique, by
which the opposition marshals contradictory cases-in this example,
persons who have no such knowledge-and
the asseverator rules them
out as "not meant" by his statement. The universe of ordinary discourse
is vague enough so that this process can go on as long as the bellicosity of
the two adversaries lasts. Logicians and scientists, however, take no
pleasure in casuistry. Their universe of discourse must be definite
enough to allow no dispute whatever about what does or does not belong
to it.*
The fixed universal set we shall denote by 'U. Once having decided
on the universal set 'U for a particular discussion, all other sets in that
same discussion must be subsets of 'U. But different universal sets can
be used for different discussions. 'U may be a set of people in one
problem, a different set of people in another, a set of numbers in yet
another, etc.
* Susanne K. Langer, An Introduction
Publications, Inc., 1953, p. 68.
Sec. 3 / OPERATIONS
17
ON SETS
We now define the three basic operations on sets.
Definition
3.1. Let A and B be any subsets of a universal Bet <It.
Then
I. The complement of A (with respect to ell) is the set of elements
of'lL that do not belong to A. The complement of A is denoted by A',
the particular universal set being understood from the context. In
symbols,
A = {x e -u I x Ii A}.
I
II. The intersection of A and B is the set of elements that belong
to both A and B. The intersection of A and B is denoted by A () B,
whieh is read "11 cap B" or "A intersection B." In symbols,
A
nB
= {x
Ix
eA
and x € B}.
III. The union of il and B is the set of elements that belong to at
least one of the sets A and B, i.e., to 11 01' B. The union of A and B is
denoted by A U B, which is read "il cup B" or "11 union B." In
symbols,
A. U B = {x I x t A or x E B} .
A comment about the meaning of the word "or" in mathematics is
in order here. This logical connective is ambiguous in everyday language, sometimes being used in the inclusive sense (in which "p or q"
is taken to mean "v or q, or both p and q") and other times being
used in the exclusive sense (in which "» or q" means "p or q, but not
both"). As we have explicitly indicated by our wording, the "or" in
the definition of union of two sets is to be taken in the inclusive sense,
in which it is synonomous with the legal use of "and/or." We adhere
to the accepted mathematical usage and shall henceforth always so
interpret the word "or." The words "not," "and," and "or" are
italicized in Definition 3.1, for they are the key words to remember
in the definitions of complement, intersection, and union of sets.
The following examples illustrate how one obtains new sets by
applying the operations in Definition 3.1 to given sets.
Example 3.1. Let the universal set 'U be the set of letters in the
alphabet, and let 11 be the subset of vowels, and B the subset containing the first three letters, i.e.,
to Symbolic Logic, 2nd edition, Dover
A = {a, e, i,
0,
10},
B
=
{a, b, c}.
__
. .. _ .. .._ ....•.... _
- ~..".-.~_--.-..-.......,.
I
.• 1
SETS / Chap. 1
22
It will be helpful in our later work to have clearly in mind the correspondence between various subsets of 91 and regions of the Venn
diagram in Figure 7. The reader should check the examples in
Table 3.
tively, then all people are classified into one of the eight categories
indicated in the Venn diagram of Figure 8.
Suppose a laboratory technician reports the following statistics
after testing blood samples of 100 people:
TABLE 3
Set
RI & R2 & R3 & R, & R5 & Rs & R7 & R8
RI&R2&R3&R.
RI & Rz & R5 & Rs
RI & R3 & Rs & R1
RI&R2
e
AnB
AUB
(A U B)
ne
A'
(A n B)
Bne
(A n B') n c:
A'
l
n
50 contain
contain
40 contain
20 contain
13 contain
15 contain
5 contain
52
Region in Figure 7
'lL
A
B
RI & R2 & R, & R, & R5 & RG
RI & R3 & Rs
R5& R6 & R7& Rs
None (the set is empty)
RI&R5
R,
-
-
We see that by starting with two or more subsets of some universal
set and forming their complements, unions, and intersections, many
other subsets are obtained. In the next section, we explore certain
interesting and important relationships among these subsets. We
conclude here with another example in
which a Venn diagram proves helpful.
E)(dfllpl e 3.5. Persons
are classified
according to blood type and Rh quality
by testing a blood sample for the presence of three antigens: A, B, and Rh.
Blood is of type AB if it contains both
antigens A and B, of type A if it contains
A but not B, of type B if it contains B
but not A, and of type 0 if it contains
Figure 8
neither A nor B. In addition, blood is
classified as Rh positive (+-) if the Rh antigen is present, as Rh
negative (--) otherwise. If we let A, B, and Rh denote the sets
of people whose blood contains the A, B, and Rh antigens respec-
23
Sec. 3 / OPERATIONS ON SETS
antigen A
antigen B
antigen Rh
both A and B
both A and Rh
both Band Rh
all three antigens
How many persons of type A - did the technician find? To answer this sort of question, we use the data to fill in the number of
people in each of the eight subsets in Figure 8. The trick here is to
use the data in reverse order, i.e., work from the bottom of the list
to the top. Thus, the last item reported tells us there are five people
of type AB+-. The 15 people reported to have both Band Rh must
be of typeAB+ or B+. Since five people are already identified as
type AB+, we infer that ten people are of type B+-. In this way we
complete. the enumeration, and thus obtain the number of people in
each of the eight categories. We find there were 22 people of type
A-.
PROBLEMS
3.1. Let 91= {a, b, c}, A = {a}, B = {b}. List the elements of the following sets: A', B', A U B, A r, B, A' r, B', A' r, (A U B).
3.2. Refer to the Venn diagram in Figure 6. Determine the region or combination of regions representing each of the sets (a) (A U B)',
(b) A' U E',
(c)
A U B',
(d)
(A')',
(e)
(A'
r, B)' U
B.
3.3. A universal set 91 has eight elements corresponding to the eight possible outcomes of the experiment in which a penny, a nickel, and a
dime are tossed.
(a) List the elements of 91.
(b) Suppose subset A contains those elements corresponding to outcomes for which the penny falls heads, subset B those for whichall
three coins match, and subset C those for which the number of
heads exceeds the number of tails. List the elements of the following sets: A', B', A V B, A V C, B V e, A r, B, A r, C, B n C,
A' r, C, (A r, B) r. C, and (A r, B') r. c.