Homer Walker , lurie, Zou, HILL, Malone, JohnGoulet, Doytchinova, Volkov, Rostami, PeterChristopher, Wu, jacob, Gagnon, LIU, Jon Abraham, William Martin
MA 2073, C Term 2018, HW #2
Instructor: L. Carichino
Due: 1/22 in class.
Name:
Total points: 20
Homework guidelines:
(1) Students can collaborate on the homework, but are expected to submit individual
MA 2073, C Term 2018, HW #1
Name:
Instructor: L. Carichino
Due: 1/16 in class.
Total points: 20
Homework guidelines:
(1) Students can collaborate on the homework, but are expected to submit individual
Introduction To Statistics for Life Sciences
Zheyang Wu
October 22, 2017
1
Statistics and Life Sciences
Why Statistics?
Data from researches in the life sciences (e.g., clinic, laboratory, greenhouse
Ma2201/CS2022
Final Exam
Discrete Mathematics
D Term, 2013
SOLUTIONS:
ENJOY YOUR SUMMER!
1. (10 points) Let A = cfw_1, 2, 3, . . . , 10, let B = cfw_1, 2, 3, . . . , 210 and let C = P(A) be
the power
Ma2201/CS2022
Quiz 0001
Discrete Mathematics
D Term, 2013
Solutions
1. (3 points) You place 13 cards from an ordinary 52 card deck in order on a table.
How many ways are there to do this.
52!
52!
a)
c
Ma2201/CS2022
Quiz 0110
Discrete Mathematics
D Term, 2013
Solutions
DOUBLE DOWN QUIZ:
Normal
Double Down
TRIPLE DOWN
1. (4 pts) How many 144 letter strings of lower case letters L = cfw_a, b, c, .
Ma2201/CS2022
Quiz 0010
Discrete Mathematics
D Term, 2013
Solutions
1. (4 points) Let A, B, and C be nonempty subsets of X = cfw_1, 2, 3, . . . , 100. Suppose
A = cfw_1, 2, 3, . . . , 50 = cfw_n X | n
Exercises: Pascals Triangle
Some of these exercises use the Hockey Stick Identities
(
)
) (
)
n ( )
n (
i
n+1
i
n+1
=
,
=
k
k+1
ik
nk
i=k
i=k
1. Show by induction that the binomial
coecients are bimod
Ma2201/CS2022
Quiz 0011
Discrete Mathematics
D Term, 2013
Solutions
1. (6 points) Suppose p, q and r are statements and suppose (p r) and (r q)
are both true.
Circle each of the following which must b
Exercises: Stirlings Formula
n! (n/e)n 2n,
lim
n
n!
=1
(n/e)n 2n
1. Use Stirlings formula to estimate 2 4 6 2n.
2. Use Stirlings formula to estimate 1 3 5 (2n 1).
3. Use the previous two exercises to
Ma2201/CS2022
Quiz 0111
Discrete Mathematics
1. (2 points) Consider f (n) =
nn
. Circle the best answer.
n!
a) f (n) 3n
c) f (n) grows faster than 2n
By Stirlings formula f (n)
D Term, 2013
Solutions
Discrete Mathematics
Ma2201/CS2022
Quiz 0101
D Term, 2013
Solutions
1. (2 points) Let X and Y be finite sets. What would allow us to conclude that every
one-to-one function with domain set X and targe
Discrete Mathematics
Ma2201/CS2022
Quiz 0100
D Term, 2013
Solutions
1. (4 points) Suppose that cfw_pn is a set of statements for n cfw_1, 2, 3, . . ., and suppose
that for all n we have pn = pn+1 .
S
Discrete Mathematics
Ma2201/CS2022
Final Exam - at least for this class.
A Term, 2016
Print Name:
Sign:
Do any six of the following eight problems:
1. (5 pts) Prove by induction for all n 1 that 1 + 3
Ma2201/CS2022
Final Exam
Good Luck
Discrete Mathematics
D Term, 2011
Print Name:
Sign:
Choose any five of the following.
1. (6 pts) How many strings of six characters either start and end with the sam
Discrete Mathematics
Ma2201/CS2022
Final Exam - at least for this class.
d Term, 2017
Print Name:
Sign:
Do any six of the following eight problems:
1. (5 pts) Prove by induction that for all n 1 that
Ma2201/CS2022
Quiz 0101
Discrete Mathematics
A Term, 2016
Print Name:
Sign:
1. (2 pts) Let A, B, and C be sets and let A B C. Circle the item below which
must be true.
a. |A| |B C|
c. If A is countabl
Discrete Mathematics
Ma2201/CS2022
Quiz 0010
A Term, 2016
Print Name:
Sign:
1. (6 pts) Let p, q and r be statements. Show by any method we have learned that
(p r) (q r) (p q r)
Method 1: You can make
Ma2201/CS2022
Quiz 0011
Discrete Mathematics
A Term, 2016
Print Name:
Sign:
1. (4 pts) Let p be the statement I have slept five hours. Let q be the statement I
have a hot cup of coffee. Let r be the s
Ma2201/CS2022
Quiz 0110
Discrete Mathematics
A Term, 2016
Print Name:
Sign:
1. (3 pts) Find the multiplicative inverse of 23 in Z101 .
The obvious ones in Z101 come from 1 102 = (2)(51) mod 101 and 1
Ma2201/CS2022
Quiz 0001
Discrete Mathematics
A Term, 2016
Print Name:
Sign:
1. (10 pts) Let p, q and r be statements. Let p be TRUE and q be FALSE.
Label each of the following as TRUE, FALSE, or, if i
Ma2201/CS2022
Quiz 0111
Ordinary Quiz
Discrete Mathematics
A Term, 2016
Print Name:
Sign:
Double Down
Triple Down
1. (2 pts) Find the multiplicative inverse of 25 in Z103 or show that one does not exi
Ma2201/CS2022
Quiz 0100
Discrete Mathematics
A Term, 2016
Print Name:
Sign:
1. (6 pts) Let A, B, and C be sets.
Prove using the double inclusion method that A (B C) = (A B) (A C).
We first show A (B C
We discussed relations and functions. A relation R from A to B is a subset
R A B. A relation on A A can be reflexive, symmetric or transitive.
A relation which is reflexive, symmetric and transitive i
Summary for lectures 3 and 4
We are studying sets.
We defined sets, and the notation for sets, and gave several examples.
We discussed the notion of a set being well-defined.
Introduced the notation a
Lectures 12 and 13
We started formal logic which concerns statements. A statement is either TRUE
(1) or FALSE (0).
Statements can be formed from (AND), (OR) and (NOT).
We stated the distributive laws:
In lecture 7 we finished the material on lexicographic order, and Russels
Paradox.
In honor or Labor Day, there is just one problem.
1. Let A = cfw_a, b, c, d, e,B = cfw_a,c,e,g,i, and C = cfw_d, e, f
In lectures 5 and 6 we continued to discuss sets.
We discussed the double inclusion method of showing two sets are equal.
We discussed the cartesian products of sets and Lexicographic order.
We discus
Exercises for Lectures 10 and 11
These lectures considered functions and cardinality.
|A| < |B|: There is a one-to-one function from A to B.
or
There is an onto function from B to A.
|A| = |B|: |A| |B
Exercises for Lecture 2
Today we discuss the binary numbers, and number bases in general.
Binary Arithmetic
1. Covert these decimal numbers to binary: 2017, 24816, 11111.
2. Convert these binary numbe