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 + 5sn-2 for n 2
s0 = 4, s1 = -20, and sn = -4sn-1 + 5s
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 + . + 1/(n -1)n = (n -1)/n
Proof (by induction) For each
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. The same result(s) hold
for any prime k.
Theorem. (This
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 in class (especially
on Wed., April 3), they all mean e
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 and NEATLY below.
Closed Book
Closed Calculator
Focused E
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 for p(n), ~half
for n -15); at this
point, any domain tha
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
your solutions to be turned in.
Due Friday, Feb 6:
Be s
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 such that ax = b - 3.
SOLUTION.
FIRST, rewrite the fact to
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 proof amounted to saying that because, for all x 0, f
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 solutions.
(Sometimes there will be, sometimes not.)
As
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 that f is one-to-one iff either of the following equiva
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 changed, to protect the innocent.
1. Suppose you have a func
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. Find an integer m such that the inequality is true fo
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) = |x |. Otherwise (for other real numbers), h (x) = 0.
(
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 carefully and completely that ~ is an equivalence relati
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, there exists x < 3 such that y = f (x).
SOLUTION. Suppo
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 let W be the set of all points on the white board
at th
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 specific, simple function from W to Q, the set of
all ration
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 solution.
Consider a function f : (-!, 3) ! [2, !) defi
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 changed.
1. Suppose that R is a relation on a set A. Supp
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 RESULT to prove:
If m is an even number and n is an integer,
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);
there are 10 possibilities for each of the second, third, a
Math 243-1
Final Exam
Solutions to Selected Problems (6 and 8)
2012
6. Think of the length function on the set * of words formed from an alphabet . State the image
of the length function and explain your answer carefully.
SOLUTION. If we assume, as usual,
Math 243-1
Final Exam Solutions to Selected Problems (10)
2012
10. Define the relation R on the set of real numbers by m R n iff m - n is an integer which is divisible by 7. (SOLUTION to part (a) is below; this is an equivalence relation on the set of rea
Math 243-1
Final Exam Solutions to Selected Problems (6 and 8)
2012
6. Think of the length function on the set * of words formed from an alphabet . State the image of the length function and explain your answer carefully. SOLUTION. If we assume, as usual,
Math 243-1
Exam 3, SOLUTIONS, Problem 6
2012
6. Find an explicit general formula for sn , where s1 = 1, s2 = 1, and sn = sn -1 + sn -2 for n 3. You may already know the solution, but solve the equation in the standard way as if you didn't already know the
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-defined function. SOLUTION. By Problem 3, it doesn't matter w
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 class for doing induction proofs. SOLUTION. For each posit