MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #1.
1. Construct the truth-table for the following propositional formulas. In each case, explain
whether the formula is a tautology
MAT2362: Foundations of Mathematics
Homework Assignment #2.
Instructions.
This assignment should be handed in at the beginning of class on Oct.
10th. If you hand in your assignment late(r), I will ma
MAT2362: Foundations of Mathematics
Homework Assignment #3.
Instructions.
This assignment should be handed in before 4pm on Dec. 4th. You can
hand in your assignment during the lecture or DGD, or at
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #4.
1. [3 points] Suppose that f : R Z is a function. Prove that at least one of
the bres of f must be innite.
Solution. It is usef
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #3.
1. [6 points] Let f : X Y and g : Y Z be functions, and suppose g f is bijective. For each of the following, either give a proo
MAT2362: Foundations of Mathematics
Homework Assignment #2 Solutions.
1. [4 points] Show that for any sets A, B, C, we have
A\(B\C) = (A\B) (A C)
If you appeal to facts learned in class, indicate clea
MAT2362: Foundations of Mathematics
Homework Assignment #5 Solutions.
1. [4 points] For each of the following subsets of R, nd the inmum and
supremum (if such exist). You dont have to motivate your an
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #2.
1. Consider the set A = cfw_a, b, c, d and consider the following relations on A:
R1
=
R2
= cfw_(a, a), (a, b), (a, c), (a, d)
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #3.
1. [ 4 points] For each of the following, give a concrete example. (Meaning:
completely specify the set and the relation.)
(a)
MAT2362: Midterm Exam #2 solutions.
Prof. P. Hofstra
November 6, 2015
1. [8 points] For each of the following statements, indicate whether it is true or false. You
dont have to justify your answers.
(
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #3.
1. [ 3 points] Consider the function f : N N given by f (x) = x3 . Find
the following sets (explaining how you arrive at your a
MAT2362 Foundations of Mathematics
Solutions to Midterm Test #2
1. [10 points] True or False? For each of the following statements, indicate whether it is
true or false. You dont have to motivate your
MAT2362: Foundations of Mathematics
Quiz #8. Submit by Monday Dec. 4, 4pm.
NAME:
.
Student ID:
.
1. Let A be a set and suppose that f : N A is a function. Which of the following is not correct?
A. If
MAT2362: Foundations of Mathematics
Quiz #8. Submit by Monday Dec. 4, 4pm.
NAME:
.
Student ID:
.
1. Let A be a set and suppose that f : N ! A is a function. Which of the following is not correct?
A. I
MAT2362: Foundations of Mathematics
Quiz #5.
NAME:
.
Student ID:
.
1. Consider the set A = cfw_0, 1, 2, 3 and the relation R defined by
R = cfw_(0, 1), (2, 1), (1, 3), (3, 2), (3, 1).
Which of the fol
MAT2362: Foundations of Mathematics
Homework Assignment #2.
Instructions.
This assignment should be handed in before 3pm on Nov. 3./ You can
hand in your assignment at my office, or send me an electr
MAT2362: Solutions to Midterm Exam #1.
Prof. P. Hofstra
October 9, 2015
1. [7 points] For each of the following statements, indicate whether it is true or false. You
dont have to justify your answers.
Definitions, Statements
Equivalence Relation
An equivalence relation on a set X is a relation R X X such that:
( x , x ) R x X (reflexive property);
( x , y ) R ( y , x) R (symmetric property);
( x ,
MAT2362: Foundations of Mathematics Mini-test #1.
Name:
Student Number:
yaran tao 8257263
Instructions.
This test is take-home, and should be submitted before noon on Monday, Sept.
18. You can hand i
Foundations and Fundamentals of Mathematics
Course material for MAT2362
Pieter Hofstra
Department of Mathematics and Statistics
University of Ottawa
Contents
Introduction
ix
Chapter I
I.1
I.2
I.3
I.4
Properties The density of air is given to be 1.18 kg/m3 at the
beginning, and 7.20 kg/m3 at the end.
Analysis We take the tank as the system, which is a control
volume since mass crosses the boundary.
Assumptions 1 This is a steady-flow process since there is no change with time. 2 Kinetic and potential
energy changes are negligible. 3 The turbine is adiabatic and thus heat transfer is negligible.
Assumptions 1 Steady operating conditions exist. 2 The heat exchanger is well-insulated so that heat loss
to the surroundings is negligible and thus heat transfer from the hot fluid is equal to the he
Assumptions 1 Steady flow conditions exist. 2 Air is an ideal gas with constant specific heats. 3 The
pressure of air is 1 atm. 4 Kinetic and potential energy changes are negligible
Properties The spe
Assumptions 1 This is a steady-flow process since there is no change with time. 2 Potential energy
changes are negligible. 3 The device is adiabatic and thus heat transfer is negligible.
Properties Fr
Assumptions 1 Steady operating conditions exist. 2 The mixing chamber is well-insulated so that heat
loss to the surroundings is negligible. 3 Changes in the kinetic and potential energies of fluid st
MAT2362: Foundations of Mathematics
Solutions to Quiz #2.
1. We consider 33 matrices (over the real numbers). We use the usual notation: aij is the entry in the ith row
and jth column. Indicate which
MAT2362: Foundations of Mathematics
Solutions to Homework Assignment #2.
1. For each of the following statements, either give a counterexample or a
short (3 lines max) explanation of why it is true.
(
MAT2362: Foundations of Mathematics
Mini-test #2.
Instructions.
Write your answers in the answer boxes. Make sure to write your name and student ID. Dont use red ink
or pencil.
Unreadable or ambiguo
MAT2362: Foundations of Mathematics Mini-test #1.
Name:
Student Number:
Instructions.
This test is take-home, and should be submitted before noon on Monday, Sept.
18. You can hand in your test by ema