Solutions MAT 300 (H. Thieme) 1 30 2 23 3 20 4 10 5 17 100 Test 3; April 21, 2006
Work your problems in the space provided. Show all work clearly. 1. Let A = {1, 2, 3} and consider the following relation R on A, R = {(1, 2), (2, 1), (3, 2)}. [30 poi
MAT 300 RECITATIONS WEEK 2 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Determine whether each of these conditional statements
is true or false.
(1) If 1 + 1 = 3, then unicorns exist.
True. Since statements 1 + 1 = 3 and unicorns exist are both
false, and c
MAT 300, Mathematical Structures Problem Set 4 Solutions
Dr. L. Mantini Spring 2010
3.1: 2, 3, 8, 12, 15, 16; 3.2: 2, 3, 7, 12; 3.3: 2, 6, 18, 19, 20, 21. (Some proofs are given in several versions.) 3.1.2 Consider the Theorem: Suppose that b2 4ac > 0. Th
MAT 300, Mathematical Structures Review for Exam 1
Spring 2010
Exam 1 will consist of approximately 60% proofs and 40% related material. The breakdown is as follows: (I) Denitions (510%). Be able to give proper mathematical denitions, as we have discussed
MAT 300, Mathematical Structures Problem Set 3 Solutions
Problems 2.1: 2, 5, 7, 8; 2.2: 2, 3, 7, 9, 10; 2.3: 3, 5, 6, 9, 11, 12
Dr. L. Mantini Spring 2010
2.1.2 Analyze the logical forms of the following statements (other answers are possible). (a) Anyone
MAT 300 RECITATIONS WEEK 10 EXERCISES
LEADING TA: HAO LIU
Exercise #1. Determine if the proofs below are legitimate proofs.
If not, identify what is wrong with it. If the original claim is true,
prove it correctly.
(1) Claim: Suppose f : A B and S and T a
MAT 300 RECITATIONS WEEK 10 EXERCISES
LEADING TA: HAO LIU
Exercise #1. Complete the proof of the following claim:
Claim: Suppose that f : X Y is a function. If A X and B X,
then f (A B) = f (A) f (B).
Proof. () For any y U ,
y f (A B) x A B such that y =
MAT 300 RECITATIONS WEEK 9 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Let F = cfw_An nN be any family of sets. Let X be any
subset of U . Represent the following statements using , , , , ,
or , but without using P, , and cfw_, .
(1) X P( An ).
n=1
Solutio
MAT 300 RECITATIONS WEEK 8 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Represent the following sets without using , , then
give your proof.
(1)
100
n=1
0, 1
We claim
1
n
100
n=1
0, 1
99
1
= 0,
.
n
100
Proof :
1
99
0,
.
n
100
1
Let x 100 [0, 1 n ] be arb
MAT 300 RECITATIONS WEEK 7 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Use induction to prove that for every natural number
n 4, n! > 2n .
Proof. For any n N with n 4, let P (n) be the statement n! > 2n .
(i) For n = 4,
4! = 24 > 16 = 24 = 2n
.
(ii) Let n
MAT 300 RECITATIONS WEEK 6 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Prove or disprove: For any sets A, B, C, and D, if A C
and B D, then A B C D.
Proof: Let A, B, C and D be arbitrary sets. Suppose A C and
B D. Let x U be arbitrary. Suppose x A B. Then
MAT 300 RECITATIONS WEEK 5 SOLUTIONS
LEADING TA: HAO LIU
Note: Let U be the universe of discourse.
Exercise #1. Prove or disprove the following statements involving
universal quantiers.
(a) a U, b U, if (a Q) (b Q), then (a + b) Q.
/
/
Proof by contradict
MAT 300 RECITATIONS WEEK 4 EXERCISES
LEADING TA: AVIVA HALANI
Exercise #1. Write the negation of each statement. It may help to
rewrite the statement in a form that is more familiar to you.
(1) If it snows at night, then I will stay home.
Negation: There
MAT 300 RECITATIONS WEEK 1 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1.
Part 1.
(1) r p.
(2) (p q) r.
(3) (q p) r.
Part 2.
(1) It is not the case that both grizzly bears have been seen in the
area and hiking is safe on the trail.
(2) It is not the case that
MAT 300 RECITATIONS WEEK 14 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Prove Theorem 6.2.4(ii). Let X, Y, Z, W be any sets
with X Y and Z W . If X Z = = Y W then X Z Y W .
Proof. Suppose X Y , Z W and X Z = = Y W . Then there
exist bijections f : X Y and
MAT 300 RECITATIONS WEEK 13 SOLUTIONS
LEADING TA: HAO LIU
Exercise #1. Let f : (0, ) (0, 1) dened by f (x) =
that f (x) is a bijection. Find f 1 and give a proof.
x
. Show
x+1
Proof. First well show f (x) is a bijection.
(1) We rst show f is injective.
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B g D, then,4 U B g C U D.
and
Fill in the blanks for the complete proof:
proof Assume ArB,C
By definitior of
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MAT 300 LIST OF DEFINITIONS
Section 1.1:
A proposition is a sentence that is either true or false.
An argument is a sequence of statements, called premises followed by a statement called the
conclusion.
An argument is valid if and only if given that all p
MAT300 Project1
Zhiheng Wang
1.
1). Counterexample:
Let domain D = cfw_1, 2, and let P 1) = true, P(2) = false, Q(1) = false, Q(2) = true.
Then xP(x) xQ(x) is false, but x (P(x) Q(x) is true.
2).
xP(x) xQ(x) => xy(P(x) Q(y)
Suppose xP(x) xQ(x),
Case1: xP(
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MAT 300
Professor Spielberg
Fundamental Distributive Laws
In quantificational logic there are two main quantifiers, which come in the
symbolic form of , and . The first symbol is known as the universal quantifier that
sta