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 mark your assignment
but there is a penalty.
Make sure y
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 useful (also for problem 5) to prove the following more gen
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 proof or a
counterexample:
(a) f is injective
Solution.
If
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 clearly which ones, and
how you use them.
Solution. You can
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 answer.
1
(a) A = cfw_ n + 2|n N+ .
Solution: note that t
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)
R3
= cfw_(a, b), (b, c), (c, a), (d, a)
R4
= cfw_(a, b
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) A relation R on a set A which is reexive, symmetric and
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.
(a) For any sets A and B, we have A B = B A.
Solution. F
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.
(a) Consider an argument with premises p1 and p2 and c
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 of the matrices below satisfy the following condition (
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, a contradiction, or neither. (Explain how you arrive
Quiz 1 Solutions
1. Consider the following propositional formulas:
= (q (q p) q;
= q (q p) q).
Which of the following statements is true?
A. Both and are tautologies.
B. is a tautology and is a contingency.
C. is a contingency and is a tautology.
D. and
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. The mass balance for
this system can be expressed as
M
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.
Properties From the steam tables (Table A-6)
P1
12.5 MP
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 heat transfer to
the cold fluid. 3 Changes in the kinetic
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 specific heat of air at room temperature is cp = 1.005 kJ/
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 From the steam tables (Tables A-4 through 6)
P1
T1
10 MPa
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 streams are
negligible. 4 Fluid properties are constant.
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 answer.
(a) Any reexive relation is total.
Solution. T