Math 243
Exam 3D and 3B, Part I, Solution 3
Solutions for two different versions of the exam.
Fall 2015
3. (a) Find an explicit general formula for sn , where
Exam 3D:
s0 = 4, s1 = 10, and sn = -4sn-1
Math 243
For all y in ThisSet, there exists x in ThatSet such that .
2015
Suppose A and B are sets and f is a function.
We have seen many statements of the following form, and as has been pointed out
Math 243
Simple algebra.
April 3, 2015
Since there is room below to write your answers clearly and neatly on this sheet of paper
(sometimes there will be, sometimes not), write your answers CLEARLY an
Math 243
II.
Solutions, Induction Problems.
2015
Prove by induction that for all integers n -15,
n2 + n is even.
Proof (by induction) For each n -15, define
p(n) : n2 + n is even .
15 points
(~half fo
Math 243
Solutions, Exercise 1.3.3.
2015
The online assignment (part of it copied here for reference):
Lesson 15.
NOTE 1: Be sure to read Homework Format and Homework Writing Policy BEFORE writing up
Math 243-2,4
WORKSHEET SOLUTION
Feb 2, 2015
2. True or false or DMS [Doesnt Make Sense]; prove if true or false; explain if DMS:
Suppose a and b are real numbers.
Then there exists a real number x suc
Math 243
How Do You Prove Two Sets Equal?
Cooperation encouraged.
2015
Background.
Students are given the function f (x) = x, defined for all x 0, and are asked to prove that
Im( f ) = [0, ).
A common
In Lesson 9.2, the following problem was given:
1. In Lesson 9.1, you used (we hope) the definition of even number to prove that for every even
number n and every integer m, nm is even. USE THIS RESUL
Math 243
Solutions, Exercise 1.1.2.
2015
2. (a) How many 4-digit numbers are there?
SOLUTION. Counting from left to right in a 4-digit number,
there are 9 possibilities for the first digit (1-9);
ther
Math 243-2,4
Exam 1A Solutions, 7, 8
Spring 2015
7. Prove or disprove each of the following statements:
a. There exists a real number x such that 2x - 1 = 4.
b. There exists a negative number x such t
Math 243-2,4
Exam 1D Solutions, 3, 4, 5, 6
Spring 2015
3. For the purposes of this problem, we will use the following definitions:
We will say that a number n is happy if and only if there is an integ
Math 243
Solutions, solving equations in Z(k)
Regarding Problem 4 in Lesson 49.2,
and two of the problems in Lesson 52.
2015
Prove the following (you can assume, as in the homework, that k = 5 below.
Math 243
Solutions, Induction Problems.
2015
As usual, specific words used are not important; other words can convey the same ideas.
I.
Prove by induction that for all integers n 2,
1/1. 2 + 1/2. 3 +
Math 243
Induction with Recurrence Relations
2015
Before writing your answers to these questions, think about the following:
There is or there is not room on this sheet of paper to neatly write your s
Math 243
Exam 3B, Part I, Solution 2
2. For each positive integer n, consider the inequality 3 n -1 n!.
a. Is the inequality true when n = 1?
SOLUTION. True; when n = 1, 3 n -1 = 3 1 -1 = 1 = 1! n!.
b
Math 243
Exam 2A, Part II, Solutions 11, 12
Fall 2015
Your solutions should be legible, with explanation unless otherwise indicated.
11. Consider the function h defined on R by
For each x ! Z, h (x) =
Math 243
Exam 2A, Part II, Solutions 8, 9, 10
Fall 2015
Slightly updated 5 PM, March 28
8. Suppose f : A ! B. Let ~ be a relation on A defined by a ~ b iff a " A, b " A, and f (a) = f (b).
(a) Prove c
Math 243
Exam 2C, Part I, Solutions, Problems 3, 4, 6
Fall 2015
3. For each real number x < 3, let f (x) = |x | + 2. So f : (-!, 3) ! R. Prove directly and carefully that
for every real number y " 2,
Math 243
Exam 2C, Part I, Solutions, Problems 7
Fall 2015
7. Consider the following situation (we discussed a very similar situation in class).
Let X be the set of all students in this classroom, and
Math 243
Exam 2C, Part I, Solutions, Problems 0, 1
Fall 2015
0. (a) Let W be the set of all points on the white board at the front of this room.
Define (i.e., construct, give an example of . ) a speci
Math 243-2,4
Onto (see Lessons 33 and 33a)
Corrected March 8, 2015
2015
For an algebraic function like f in this problem, proving onto is equivalent to proving that a
certain algebraic equation has a
Math 243
One-to-one and Onto
2015
The instructions for each of the following problems are similar, it terms of what youre supposed to
write on the paper. Only the names and the properties have been ch
Math 243
One-to-one and Onto
2015
The instructions for each of the following problems are essentially the same; after you have read the
first, the rest are very similar. Only the names have been chang
Math 243-2,4
One-to-one and Onto
2015
Suppose f is a function from a set S to a set T, f : S ! T.
One-to-one. (Note that the following definitions are independent of the choice of codomain T.)
We say
Math 243-2,4
Exam 1A Solutions, 1, 2, 5
Spring 2015
1. On this exam, the floor of a real number x is denoted by x and the ceiling of x by x . You are,
of course, welcome to use the usual notation for
Math 243 Exam 2 November 4, 2016
Name:
INSTRUCTIONS: READ BEFORE BEGINNING
1. DO NOT OPEN THIS EXAM UNTIL YOU ARE TOLD TO DO SO .
2. There are 8 questions on the exam, not all are of equal difculty.
Math 243-1
Exam 3, SOLUTIONS, Problems 1-3, 5a
2012
1. Use the Principle of Mathematical Induction to prove that, for each positive integer n,
p =1 (2 p 1)
n
= n2.
Follow the guidelines given in clas
Math 243-1
Exam 3, SOLUTIONS, Problems 4, 5, 7, 8
2012
4. Define a function sq from [Z]3 to [Z]3 (the set of equivalence classes mod 3) by
sq ([n ]3) = [n 2 ]3.
Explain clearly why this is a well-defi