Download CSE 1400 Applied Discrete Mathematics Fall 2016 Practice Quiz 4

Name:
CSE 1400
Fall 2016
1
Applied Discrete Mathematics
Practice Quiz 4
Counting
1. (10 pts) Let X = {0, 1, 2, 3, 4} and Y = {a, b, c}.
Score
(a) How many relations can be defined between X and Y. (Hint: Recall, a relation can be represented
as an adjacency matrix. Also, a relation between X and Y is a subset of the Cartesian product
X × Y.)
(b) How many functions can be defined between X and Y. (Hint: Recall, a function can be represented
as an adjacency matrix with only one 1 per row. Also, a function from X and Y is a subset of the
Cartesian product X × Y where each x ∈ X in a pair occurs exactly once.)
Score
2. (15 pts) Dr. Cooper asked his assistant Ms. Jensen complete several tasks.
(a) List all relations from Z7 to Z9 , She politely refused. How many relations from Z7 to Z9 are
there?
(b) “Well then,” Dr. Cooper said, “give me all functions from Z7 to Z9 !” To which Ms. Jensen said,
“Well maybe that is feasible. Let me calculate.” How many functions are there from Z7 to Z9 ?
(c) “Still too many?” Dr. Cooper smirked, “list all subsets of Z9 .” How many subsets are there of
Z9 ?
2
Score
Relations
1. (15 pts) Consider the following relations between {0, 1, 2} and {a, b, c}. They are represented as
adjacency matrices. Which are reflexive? Which are symmetric? Which are antisymmetric?
a b
0 1 0
(a) 1  1 0
2 0 0

Score
c

1
0
0
a b
0 1 0
(b) 1  1 1
2 1 0

c

0
0
1
a b
0 1 1
(c) 1  1 0
2 1 0

c

1
0
1
2. (10 pts) Let a and m be integers. Write a ⊥ m if 1 is the only integer that divides both a and m.
Another way to say this is: The greatest common divisor gcd(a, m) is 1.
If a ⊥ m, say a and m are said to be relatively prime (some call it co-prime).
(a) Which is True and which is False: 12 ⊥ 3? 12 ⊥ 5? 21 ⊥ 10?
(b) Is the “relatively prime” relation reflexive, symmetric, antisymmetric, or transitive?
Score
3. (10 pts) Say two real numbers x and y are related if
and y = π are related because
equivalence.
x
y
=
π/2
π
x
y
is a rational number. As an example, x = π/2
= 1/2 is a rational number. Prove that this relation is an
1
3
Euclidean Algorithm and Linear Congruence Equations
1. (15 pts) Consider the integers 19 and 43.
Score
(a) Use the Euclidean algorithm to compute the greatest common divisor of 19 and 43.
(b) Use the quotients from problem 1a to construct a magic table that computes integers t and s such
that 19t + 43s = 1.
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(c) (5 pts) Use problem 1b to determine 19−1 mod 43.
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(d) (5 pts) Use problem 1c to solve the linear congruence equation 19x = 2 mod 43.
4
Score
Binomial coefficients
1. (5 pts) What is the value of the sum of the n-th row in Pascal’s triangle? That is, what is a simple
expression of the sum
n X
n
?
k
k=0
Score
2. (10 pts) Use mathematical induction to prove the statement: “the (partial) sum of a column in Pascal’s
triangle is the value one row down and one column over.” That is,
n X
k
n+1
=
m
m+1
k=m
(Hint: The induction is on n. The value of m is fixed and k is a dummy variable of summation.)
Total Points: 100
Friday, November 25, 2016
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