Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
List of first-order theories wikipedia , lookup
Abuse of notation wikipedia , lookup
History of the function concept wikipedia , lookup
Location arithmetic wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
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. Score (c) (5 pts) Use problem 1b to determine 19−1 mod 43. Score (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 2