SOLUTIONS QUIZ 2 - MATH 2000
January 29, 2009
For the open sentence P (A) : A cfw_4, 5, 6 over the domain S = P(cfw_3, 4, 5),
(a). all A S for which P (A) is true;
(b). all A S for which P (A) is false;
(c). all A S for which A cfw_4
SOLUTIONS QUIZ 6 - MATH 2000
Let x Z. If 3 - x, then 3 | x2 1.
Proof. We give a direct proof. Since 3 - x it follows that either x = 3k + 1
for some k Z or x = 3k + 2 for some k Z.
Case 1. x = 3k + 1 for some k Z.
x2 1 = (3k + 1)2 1 = 9k 2
SOLUTIONS QUIZ 5 - MATH 2000
Let x R. Prove that if x3 5x 1 0, then (x 2)(x 4) 2.
Proof. We let
P (x) : x3 5x 1 0
Q(x) : (x 2)(x 4) 2
We give a trivial proof by proving that Q(x) is true for all x R. Note that
(x 2)(x 4) = x2 6x + 8
= x2 6x + 9
Solutions to Homework 10 - Math 2000
During the tutorial Rex solved 6.4,6.8,6.14,6.18, and 6.20.
All other solutions except 6.22 may be found in the book.
(# 6.22) Prove that if A1 , A2 , . . . , An are any n 2 sets, then
A1 A2 An = A1 A2 An .
SOLUTIONS QUIZ 10 - MATH 2000
A relation R is defined on Z by xRy if x + 5y is even. Prove R is an equivalence relation.
This is very similar to Result 8.1 page 180 of the book. The only difference
is that the 5 here replaces the 3 in the book.
SOLUTIONS QUIZ 8 - MATH 2000
Prove that 3 is irrational.
(You may use the following fact: a Z if 3 | a2 , then 3 | a. )
Solution. Please see Homework 9 solutions for the answer to this question.
Prove that the product of an irrationa
SOLUTIONS QUIZ 4 - MATH 2000
For statements P , Q, and R, list the 8 fundamental logical equivalences.
There are 2 commutative laws, 2 associative laws, 2 distributive laws, and
2 De Morgans laws.
Solution. Please consult the book page 49, Theo
SOLUTIONS QUIZ 3 - MATH 2000
For statements P and Q, show that ( Q) = (P ( P ) and Q are
Q Q P P ( P ) Q = (P ( P )
Give the definition of a contradiction
Due Friday, March 10, 2017, 11:55pm
You must refer to the executable in $L/sample to create this program
1. For inferior goods
a. The substitution and income effects are in the same direction.
b. The income and substitution effects are in the opposite direction.
c. The substitution effect causes us to buy more (less) of the good whose
price has risen (fallen
- A culture reference that identifies people that is neither racial nor geographic
- Mixed white and native brown color in Mexico
- All Mexicans are Hispanic, but not all Hispanic are Mexicans.
2. What is culture
- Culture consists of layers o