Math 55-2 Final Exam
Rob Bayer August 14, 2009
You have until 4:00pm to complete this test. No calculators, books, notes, or consultation with other members of the class are permitted. Your exam should have 9 pages. Unsupported or improperly supported ans
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Divisibility and Modular Arithmetic 1. Evaluate each of the following: (a) -17 mod 2 1 (b) 144 mod 7 4 (c) 199 mod 19 9 (d) -101 div 13 8 - (Not 7)! 2. What is 111 1 mod 11111111? Let's
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Divisibility and Modular Arithmetic 1. Evaluate each of the following: (a) -17 mod 2 (b) 144 mod 7 (c) 199 mod 19 (d) -101 div 13 2. What is 111 1 mod 11111111?
1000 1 s 8 1s
3. Prove th
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Sets, Functions, Cardinality 1. Determine whether each of the following are injective, surjective, both (bijective), or neither. Prove your answer (a) f : N N f (x) = x2 Injective (b) f
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Sets, Functions, Cardinality 1. Determine whether each of the following are injective, surjective, both (bijective), or neither. Prove your answer (a) f : N N f (x) = x2 (b) f : R R f (x
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Functions 1. Find the value of
3 i=1 3 j=1 (i
- j)
2. Determine whether each of the following are injections, surjections, bijections, or none of the three. (a) f : R R, f (x) = 3x - 2 (
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Instructions Introduce yourselves! Despite popular belief, math is in fact a team sport! Decide who will be your first scribe. Do the problems below, having a different person be the scr
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Instructions Introduce yourselves! Despite popular belief, math is in fact a team sport! Decide who will be your first scribe. Do the problems below, having a different person be the scr
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Instructions Introduce yourselves! Despite popular belief, math is in fact a team sport! Decide who will be your first scribe. Do the problems below, having a different person be the scr
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Instructions Introduce yourselves! Despite popular belief, math is in fact a team sport! Decide who will be your first scribe. Do the problems below, having a different person be the scr
Rob Bayer Instructions
Math 55: Math 55 Worksheet Logic and Propositions
August 30, 2010
Introduce yourselves! Despite popular belief, math is in fact a team sport! Find some blackboard space, a piece of chalk, and decide who will be your first scribe. D
Discrete Math Cheat Sheet
by Dois via cheatography.com/11428/cs/1340/
Complex Numbers
Discrete Probability & Sets & Whatever
Matrix Manipulations
j 3 = -j
Probability
AT: Transpose of A - Switch Rows with
j4 = 1
z = a + bj
Columns (R1 becomes C1, R2 becom
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Warmup 1. Solve the congruence 5x 16( mod 21) Linear Congruences and the CRT x 3 (mod 4) 1. Solve the system of congruences x 2 (mod 5) x 5 (mod 7) 2x 1 (mod 3) 2. Solve the system of co
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Warmup 1. Solve the congruence 5x 16( mod 21)x = -4 Linear Congruences and the CRT x 3 (mod 4) 1. Solve the system of congruences x 2 (mod 5) Solution: x 47 mod 140. x 5 (mod 7) 2x 1 (mo
Rob Bayer 4.1#10 (a) Find a formula for
Math 55 Homework 5 SOLUTIONS
1 1 1 + + + 12 23 n(n + 1)
n n+1
July 14, 2009
The first few terms are 1 , 2 , 3 , . so we guess 2 3 4 (b) We'll go by induction: BC: for n = 1, both sides evaluate to IH: Suppose IS:
1
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Warmup - Do these together! 1. How many ways are there to rearrange the letters of the English alphabet so that "CAT," "DOG," and "FISH" are all absent. 2. What sequence is represented b
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Warmups 1. If you roll 3 fair dice, what is the expected value of the sum of the dice? Problems Proper
July 28, 2011
1. What is the probability of getting exactly 3 heads when flipping a
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Labelled or Not! 1. The 55 students of math 55 have found a bag of 555 gold coins. How many different ways can they divide it among themselves if: (a) there are no restrictions on the di
Math 55 Worksheet
Permutations, Combinations 1. Five-card draw is played with a standard, 52-card deck. There are 13 different kinds, or ranks of cards: the numbers 2 to 10, and the face cards, jack (J), queen (Q), king (K) and ace (A). And there are four
Math 55 Worksheet
Permutations, Combinations 1. Five-card draw is played with a standard, 52-card deck. There are 13 different kinds, or ranks of cards: the numbers 2 to 10, and the face cards, jack (J), queen (Q), king (K) and ace (A). And there are four
Math 55 Worksheet
Instructions: Today you'll be working with TWO different groups. In group 1 you'll all be working on the same problem. Then we'll switch into larger groups with one ambassador from each problem. The ambassador will lead the group in writ
Math 55 Worksheet
July 19, 2011
Instructions: Today you'll be working with TWO different groups. In group 1 you'll all be working on the same problem. Then we'll switch into larger groups with one ambassador from each problem. The ambassador will lead the
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Induction 1. Prove that the sum of the first n odd numbers is n2 . Solution: Try this one on your own!
d d 2. Using the product rule dx (f (x)g(x) = f (x)g(x) + f (x)g (x), and the fact
Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009. Induction 1. Prove that the sum of the first n odd numbers is n2 . 2. Using the product rule d that dx (xn ) = nxn-1 .
d dx (f (x)g(x)
= f (x)g(x) + f (x)g (x), and the fact that
d dx x