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
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
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, 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
Articulation Activities
1. Write a complete statement of the Pythagorean Theorem. Make sure you clarify exactly which set of
mathematical objects the theorem applies to.
2. Write complete statements of the commutative and associative laws for multiplicati
Solution to Problem 2 on HW #3
Theorem: A triangle is equilateral if and only if it is equiangular.
Proof:
We want to show: If a triangle is equilateral, then it is equiangular.
Let be an equilateral triangle.
Then , = , = (, ), by definition of equilater
HW 2 Due 8-31
For each of the following, prove the statement true or false. Most of these tasks are adapted from Yopp (2017).
1. If ! > ! , then > .
2. If a parallelogram has no lines of reflective symmetry, then it is not a rectangle.
3. is an integer. I
HW 4 Due 9-14
Determine whether the following statements are true or false. Explain clearly which values the variables take on
that make the statements true or false. How many values can each variable take on (one, ten, infinitely many)?
For statements 5-
If then quartets
Consider whether the following sets of if, then statements are true or false. See if you notice any patterns
that you can articulate or justify. It may be helpful to consider the set of examples that make each statement true
and the set o
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 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 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.
Fo
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(
Determine whether you think the following statements are either true or false and explain your
reasoning. Assume you have to select either true or false for each statement.
1.
Given any integer number x, x is even or x is odd.
2.
The integer 15 is even or
IfThen True/False
Determine whether each of these statements is true or false and explain your reasoning. Try to
find any patterns for how and why one of these statements can be declared true or false.
1. If a number is a multiple of 3, then it is a multi
MAT 300 Homework 1 Due 8-24
We are only just learning to prove, so I understand that you may be unclear about what claims can be assumed
as true and what claims need to be supported by further argument. We will have to live with this ambiguity a
bit, but
HW 3 Due 9-7
In class we explored how there is a relationship between if, then statements and subset relations between
sets. In particular, an if, then is true whenever the set of objects satisfying the first part of the statement form
a subset of the set
HW 5 Due 9-26
We define an open interval as , = < < .
We define a closed interval as , = .
We may also define infinite intervals such as , = < or , = .
We will want to talk about groups of intervals. We do this using indices, which are simply counter vari
Chapter 1
Basic connectives and truth tables
Definition 1 A proposition is a statement that is either true or
false.
Logical connectives:
Negation: The negation of p, denoted by p or p is the
statement which is true when p is false and is false when p
is
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Chapter 4
Relations
Cartesian Product of Sets
Definition 1 (Kuratowskis definion of an ordered pair) An ordered pair (a, b) = cfw_a, cfw_a, b.
Then (a, b) = (b, a) if and only if a = b.
Definition 2 Let A, B be sets. Then
A B = cfw_(a, b)|a A b B.
Theorem
Chapter 3
Methods of Proof
Proving a conditional statement
1. Direct proof:
Aim is to show that the implication p q is true.
Method is to assume that p is true and argue that q is
true.
2. Indirect proof:
Aim is to show that the implication p q is true
Name Class Date
Chapter Test 3
Chapter: Acid-Base Titration and pH
PART I In the space provided, write the letter of the term or phrase that best
completes each statement or best answers each question.
i. The pH scale generally ranges from
a. 01:0 1.
b.
Chapter 5
Functions
Definition 1 A function f from A to B (f : A B) is a relation
from A to be such that for every a A there is a unique b B
such that (a, b) f .
Example 1
A = cfw_1, 2, 3, B = cfw_1, 2. Let
f1 = cfw_(1, 1), (1, 2), (2, 1), (3, 1),
f2 = c