Math 8 Homework #1 Solutions
1. Consider a mathematical theory in which a statement is any string of
letters where a letter is either a lowercase letter or a capital S. The only
axiom is: S. The logic
Math 8 Homework #2 Solutions
1. Give a direct proof of the following.
(1)
(2)
(3)
(4)
SP
P (G R)
G
P T
S T
Solution.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(G R) P,
(G R) P,
G R,
P
S
T
S T
Contrapositive of (2
Homework 1 Solutions
Kyle Chapman
October 26, 2012
9.1
True, True, False, True, False, False, True, False, True
9.2
a
D and E
b
You can only deduce that it is not D.
9.3
a
Not valid, would be D C then
Math 8 Winter Quarter 2016
Homework 2
Due: Wednesday 20 January
1. Let A be the set cfw_a, cfw_1, a, cfw_4, cfw_1, 4, 4. Which of the following statements are true and which
are false?
(a) a A.
(b)cfw
Math 8 Spring Quarter 2015
Homework 2
Due: Tuesday 14 April
1. Let A be the set cfw_a, cfw_1, a, cfw_4, cfw_1, 4, 4. Which of the following statements are true and which
are false?
(a) a A.
(b)cfw_a A
Math 8 Fall Quarter 2013
Homework 1
Solutions
1. (i) Prove that the product of two consecutive integers is an even number.
Solution: Let a, a + 1 denote a pair of consecutive integers. Suppose a is e
Math 8 Fall Quarter 2013
Homework 2
Solutions
Number Systems
1. Find geometrically on the line of real numbers the points that represent the following numbers:
(i) 2/5, (ii) 10.
Solution: For nding t
Math 8 Spring Quarter 2015
Practice Problems for Midterm
No notes or calculators are permitted on this exam. To obtain full midterm credit you must show
all work, provide complete proofs, and provide
Math 8 Homework #2
Due: Friday, April 15th in class
Instructions: I strongly recommend that you attempt all problems on
your own before consulting a classmate or myself. All assignments must be
writte
Homework 2 Solutions
Kyle Chapman
October 15, 2011
18.1
a
Seeking a contradiction, suppose 3 is rational. This means it can be written as p/q for some integers p and
q . Without loss of generality, as
Math 8 - Solutions to Home Work 2
Due: October 11, 2007
1. Every/Only. Sometimes sentences with the words only and every can be conditional statements in disguise. For example, Every even number is a
MATH 8 PRACTICE MIDTERM 2
Summer 2015
1. Prove that for every real number x, there exists a unique real number y such that x2 y = x y.
2. Let A, B, C, and D be sets.
(a) Prove that if A B and C D, the
Math 8 - Midterm 1 Solutions
October 19, 2007
1. (12 pts) Consider the proposition R
If I go surng or take a nap, then I will not go surng or I will not take a nap.
(a) (2 pts) Express this propositio
Cheat Sheet
1. An equivalence relation on a set A is a relation R that is:
(a) Reflexive: for all a A, aRa;
(b) Symmetric: For all a, b A, if aRb, then bRa;
(c) Transitive: For all a, b, c A, if aRb a
Math 8 Homework 6 Solutions
1. Let f : Z Z be defined by
(
f (n) =
n + 3 if n is odd
n 1 if n is even.
Prove that f is a bijection and give a formula for f 1 .
Solution. We could prove that f is injec
Math 8 Homework 4
Due Friday November 3rd (in class).
1. For each of the following relations, determine whether or not they are
reflexive, symmetric or transitive.
(a) Let | denote the relation on N w
I put the problem statement on the last page of this pdf.
(The title is because problem 3 was about a special case of the game Nim,
as described at https:/en.wikipedia.org/wiki/Nim .)
Base Case: If th
Math 8 Pigeon Hole Principle Exercise Solutions
1. Show that in any word a1 a2 a30 of length 30 from the alphabet
cfw_1, 2, 3, there exists an i, j with i 6= j and 1 i, j 28 such that
ai ai+1 ai+2 = a
Math 8 Assignment 7
Due: Monday, December 4th in class.
1. Prove the following statements using induction (for all natural numbers
n 1, unless otherwise indicated.)
(a) 3 + 11 + 19 + + (8n 5) = 4n2 n.
Notes on Proofs
Proof By
Contradiction
First sentence should be: Suppose for the sake of contradiction.
Make sure you keep the assumptions & negate the conclusion!
Everything in the meat of the proof
MATH 8 MIDTERM 1
Summer 2017
Due Wednesday, July 26 in class
1. An advertisement for a tennis magazine states, If Im not playing tennis, then Im watching
tennis. And if Im not watching tennis, then Im
Math 8
Final Exam
Instructor: Dr. Casteels
Time allotted: 180 minutes
Last Name:
First Name:
1. Clearly explain your answers. Write neatly!
2. If you run out of room, please use the back of the preced
Math 8
Test 1 - October 23rd
Instructor: Dr. Casteels
Time alloted: 50 minutes
Last Name:
First Name:
1. Clearly explain your answer. Write neatly!
2. If you run out of room, please use the back of th
Math 8 Homework 5 Solutions
1. Prove that if c 0 (mod a), c 0 (mod b) and gcd(a, b) = 1, then c 0
(mod ab).
Solution. Suppose that c 0 (mod a) and c 0 (mod b). Then there
exist integers q and s such t
Math 8
Final Exam
Instructor: Dr. Casteels
Time allotted: 180 minutes
Last Name:
First Name:
1. Clearly explain your answers. Write neatly!
2. If you run out of room, please use the back of the preced