1. (Inspired by Nancy Brown) Suppose that (A1 , B1 , f1 ) and (A2 , B2 , f2 ) are two functions
and that B1 = B2 and f1 = f2 . Must it be the case that A1 = A2 ? If so, then prove it. If
not, provide a counterexample.
2. (Inspired by Nancy Brown) Suppose
1. Prove that two functions (A1 , B1 , f1 ) and (A2 , B2 , f2 ) are equal if A1 = A2 = A, B1 =
B2 = B , and (x A)(f1 (x) = f2 (x). (Compare with Denition 4.1.9 on page 124.)
2. Let h(x) = x2 with domain (, 2] and codomain R, and g (x) = 6 x with domain
(2
Assume the following postulates.
P1: Either Real Analysis is dicult or not many students take Real Analysis.
Symbolically P1 becomes ? where r: Real Analysis is dicult. and s: Many students
take Real Analysis.
P2: If Math 230 is easy then Real Analysis is
1. Refer to our Math Department function from class and ll in the table below. (To t
everything in, youll probably need to make your own larger copy of the table.)
A
f (AC )
[f (A)]C
f 1 [f (A)]
Geom.
Appl.
Dyn. Sys.
Algeb.
Discrete
Untenured
Tenured
Old
Math 230
Instructor: Dr. Irl C. Bivens
Sets and Proofs
Spring, 12
Oce: Chambers 3040, 7048942317
Oce Hours: Mon: 2:304:00 (with some exceptions of 2:303:45 due to Departmental
Meetings), Tue: 2:004:00, Wed: 2:304:00, Thur: 2:004:00, Fri: 2:303:30.
Textboo
1. (Pigeon hole principle) Suppose that A and B are nite sets, that |A| > |B |, and that
f : A B is any function. Prove that there exist x1 = x2 in A such that f (x1 ) = f (x2 ).
2. Use the corollary proved at the very end of class to prove that N is an i
1. Exercise 3.7.8.
2. Suppose that R is a relation on a set S that is reexive and symmetric, but is not
transitive. That is, for all x S we have xRx and for all x and y in S if xRy then yRx.
However, the transitive property does not hold. Let R(x) = cfw_y
Theorem 3.3.9(2). Let F = cfw_A A denote a family of sets indexed by A and suppose
that B A. Then
A
A
A
B
Proof. Let x denote an arbitrary element of
A and let denote an arbitrary index in
A
B . Since B A, it follows that also belongs to A. Since x
A ,
Sally is a student at Davidson College. Assume the following postulates.
(a) If Sally takes Bio 111 then Sally also takes Math 230 if the biology lab is late in the day.
(b) Sally shouldnt take a history course if Sally takes Math 230 and is depressed abo
A proof of the validity of a deductive argument is a sequence of ws, each of which satisfy
one of the following:
(i) The w is a given (premise, hypothesis);
(ii) The w is a tautology such as p p;
(iii) The w is a previously proved result;
(iv) The w can o
Math 230
Review # 3
Spring, 12
Instructions. Work each problem in the space provided, showing all relevant work. (Unjustied answers will receive reduced credit.) The point value of each problem is indicated.
Please be as neat as possible and clearly disti
Math 230
Review # 2
Spring, 12
Instructions. Work each problem in the space provided, showing all relevant work. (Unjustied answers will receive reduced credit.) The point value of each problem is indicated.
Please be as neat as possible and clearly disti
Math 230
Review # 1
Spring, 12
Instructions. Work each problem in the space provided, showing all relevant work. (Unjustied answers will receive reduced credit.) The point value of each problem is indicated.
Please be as neat as possible and clearly disti
7. (24 points) Refer to the Sally Problem and in each part determine if the statement is
a logical conclusion. If the statement is a logical conclusion, give a proof, and if it is not,
provide a counterexample.
(a) If Sally takes Math 230 then Sally does
1. In each part nd the aw(s) in the syntax.
(a) (x)(x A (y B x > y )
(b) (x)(x B (y A y > x)
(c) (x A) > (y B )
(d) x A > B
(e) (x A) > (x B )
(f) (x A) (x A x B )
(g) (f )(f C S )
2. One proposed answer to 1.3.7 was (x)(x2 + x = 1 x < 0). Explain why thi
Theorem 5.3.14 The empty set and the real numbers are the only subsets of the real
numbers that are both open and closed.
Proof. We give a proof by contradiction. Suppose A is a subset of the real numbers that
has the following four properties: (i) A = ,
6. (18 points) Suppose a collection of disks that are black on one side and white on the
other are scattered over the surface of a table. A game is to be played in which there are
two types of moves (i) and (ii): (i) all the disks are ipped over ; (ii) (i
6. (16 points) Let f : R [1, 1] be dened by f (x) = sin x and dene a collection F
of subsets of R by F = cfw_A = f 1 (cfw_) : 1 1. (Here, f 1 (cfw_) refers to the
preimage of the singleton cfw_ under f . Dont confuse f 1 with the bijection
arcsin : [1, 1]
14. In each problem a function f : A B is given. Prove that f : A B is a bijection by
nding a function g : B A that satises the conditions (g f )(x) = x for all x in A and
(f g )(y ) = y for all y in B . Important. Be sure to verify that A is a codomain f
(Jonahs question) The sets do not have to be equinumerous. Take A to be the set of rational
numbers and B to be the set of irrational numbers.
Suppose that A is a nonempty set of real numbers that has a maximum value M . Prove
that A has a LUB that is equ
State the contrapositive version of (G2) in Denition 5.1.9.
Answer. If N is any real number such that G < N then there exists an element a A that
satises a < N .
5.1.12 It follows from Exercise 5.1.11 that the set B = cfw_a : a A is bounded from above
and
5.1.2 Suppose that A is bounded. Then there exists a positive real number M such that
|a| M for all a A. Equivalently,
M a M
(a A)
It follows that if A is bounded then A is bounded above by M and is bounded below by
M .
Conversely, suppose that A is bound
1. Prove that countability is preserved under the equivalence relation equinumerous. That
is, prove that if A is equinumerous with B and A is countable then so is B (and of the same
type nite or countably innite). (Hint: Theorem 4.6.5 takes care of the ca
1. (Pigeon hole principle) Suppose that A and B are nite sets, that |A| > |B |, and that
f : A B is any function. Prove that there exist x1 = x2 in A such that f (x1 ) = f (x2 ).
Proof #1. Suppose that the conclusion does not hold. Then f : A B is 11. Let
4.6.7 Assume rst that both A and B are nite sets of cardinalities m and n respectively.
By hypothesis, there exist bijections f1 : Nm A and g : Nn B . In addition, it is
m
easy to see that the translation T : Nn Nn dened by T (x) = x + m is a bijection
m
1. Assume that f : Nn +1 Nm+1 is 11, that f (n + 1) = k for some 1 k m and that
f (l) = m + 1 for some 1 l n . Dene g : Nn Nm
f (x) , x = l
g (x) =
k, x=l
Prove that Nm is indeed a codomain for g .
Proof. Let x denote an arbitrary element of Nn . There a
14. In each problem a function f : A B is given. Prove that f : A B is a bijection by
nding a function g : B A that satises the conditions (g f )(x) = x for all x in A and
(f g )(y ) = y for all y in B . Important. Be sure to verify that A is a codomain f
4.4.4 (Q2): This is false. Let A = cfw_1, B = cfw_2, 3, C = cfw_4 and dene f : A B by
f (1) = 2, and g : B C by g (2) = g (3) = 4. Then f and g f are 11, but g is not 11.
(Q3): This is true, and remains true even if we remove the condition that g be 11. A