MATH 3200 THIRD MIDTERM EXAM
Directions: Do all ve problems. No calculators are permitted.
1) Let f : X Y and g : Y Z be functions. Prove or disprove:
a) If f is injective and g is injective, then g f is injective.
b) If f is surjective and g is surjectiv
Math 3200 HW #1 Sketches of some solutions
1.2 (a) A = cfw_x S : x > 0
(b) B = cfw_x S : x 0
(c) C = cfw_x S : x < 0
(d) D = cfw_x S : |x| > 1
1.6 (a) A = cfw_. . . , 3, 1, 1, 3, . . .
(b) B = cfw_. . . , 4, 0, 4, 8, . . .
(c) C = cfw_. . . , 5, 2, 1, 4,
Math 3200 HW #2 Sketches of some solutions
2.2 (a) True.
(b) False. The elements of D are pretty clearly the Fibonacci numbers, and 33 is not a
Fibonacci number.
(c) False. The elements of A are the positive integers of the form 3n + 1, and 22 = 3(7) + 1,
Math 3200 HW #4 Sketches of some solutions
3.16 Let x Z. Prove that is 7x + 5 is odd, then x is even.
Proof. I will prove the contrapositive of the above statement, so the goal is to prove that if
x Z is odd, then 7x + 5 is even.
To that end, suppose x Z
Math 3200 HW #3 Sketches of some solutions
2.40 (a) False since P (1, 1) is true and Q(1, 1) is false.
(b) True since P (3, 4) and Q(3, 4) are both false.
(c) True since P (5, 5) and Q(5, 5) are both true.
2.46 Heres the truth table:
P
T
T
F
F
(P Q)
T
T
T
Math 3200 HW #5 Solutions
4.2 Let a, b Z, where a = 0 and b = 0. Prove that if a | b and b | a, then a = b or a = b.
Proof. Suppose a, b Z so that a | b and b | a. Then we can write a = kb and b = a for some
k, Z. Therefore,
a = kb = k(a) = ka,
so, after
Math 3200 HW #6 Solutions
4.56 Let A, B and C be sets. Prove that (A B ) (A C ) = A (B C )
Proof. First, suppose x (A B ) (A C ). Then x A B or x A C . Either way, we
know that x A. Now, if x A B , then x B ; since B C B , this means x B C .
/
/
Likewise,
Math 3200 HW #7 Solutions
5.32 Prove that there exist no positive integers m and n for which m2 + m + 1 = n2 .
Proof. Suppose, for the sake of proving a contradiction, that there do exist m, n N so that
m2 + n 2 + 1 = n 2 .
Now, notice that
(m + 1)2 = m2
Math 3200 Exam #1 Practice Problem Solutions
1. Let S = cfw_1, 2, 3 and let A = cfw_1. What is the cardinality of the set P (S ) P (A)?
Answer: By denition,
P (S ) = cfw_, cfw_1, cfw_2, cfw_3, cfw_1, 2, cfw_1, 3, cfw_2, 3, cfw_1, 2, 3
and
P (A) = cfw_, cf
Math 2250 Exam #1 Solutions
1. Are the statements P ( Q) and (P Q) logically equivalent? Explain your answer (either by
demonstrating that the statements are logically equivalent or explaining why they are not).
Answer: Yes, the two statements are logical
MATH 3200 SECOND MIDTERM EXAM
Directions: Do all ve problems. Always justify your reasoning completely. No
calculators are permitted (nor would they be helpful in any way that I can see).
1) Let N0 be an integer, and let P (n) be an open sentence whose do
MATH 3200 SECOND MIDTERM EXAM
Directions: Do all ve problems. Always justify your reasoning completely. No
calculators are permitted (nor would they be helpful in any way that I can see).
1) a) State the principle of mathematical induction.
Solution: Let
MATH 3200 PRACTICE PROBLEMS 1
PETE L. CLARK
In all of the following questions, let x, y, z be objects and A, B, C be sets.
1) Let A and B be sets.
a) What is the meaning of A = B ? Of A B ? Of A
B?
Solution: A = B means that the sets A and B have exactly
PROF. CLARKS MATH 3200 FALL 2009 MIDTERM 1
Part I: Do all of the following problems.
I.1) a) Show that the following are logically equivalent:
(i) A = (B C );
(ii) (A B ) = C .
b) Show that for all x Z, x(x2 + 1) is even.
I.2) Negate the following stateme
MATH 3200 THIRD MIDTERM EXAM
Directions: Do any four of the ve problems. If there is any doubt as to which
four problems you want me to grade, I will grade the rst four problems, whether
that is to your benet or not. Always justify your reasoning complete
MATH 3200 MIDTERM EXAM 1, WITH SOLUTIONS
Instructions: This is a closed book exam with calculators not permitted. You will
have 55 minutes to complete the exam. The extra credit problem is truly dicult,
so that it would be a poor strategy to try to solve
MATH 3200 PRACTICE PROBLEMS 1
PETE L. CLARK
In all of the following questions, let x, y, z be objects and A, B, C be sets.
1) Let A and B be sets.
a) What is the meaning of A = B ? Of A B ? Of A B ?
b) What is the meaning of A B ? Of A B ? Of A \ B ?
2) D
REVIEW FOR THIRD 3200 MIDTERM: 2009 EDITION
PETE L. CLARK
1) Show that for all integers n 2 we have
13 + . . . + (n 1)3 <
14
n < 13 + . . . + n3 .
4
Solution: Scroll to the bottom of
http:/en.wikibooks.org/wiki/Algebra/Proofs/Exercises#ix.29.
2) The distr
REVIEW PROBLEMS FOR SECOND 3200 MIDTERM
PETE L. CLARK
1)a) State Euclids Lemma (the one involving prime numbers and divisibility).
b) Use Euclids Lemma to show that 31/5 and 51/3 are both irrational.
2) Let x, y Z, and suppose that x is of the form 9k + 3
3200 PROBLEM SET 8: TYPED PROBLEMS
Recall that a relation R between sets X and Y is, formally, given by a subset R
of the Cartesian product X Y . In many interesting cases we have X = Y , and
then instead of saying a relation between X and X we generally