Discrete Math Final Exam

What is the coefficient of x^3y^4 in (-3x + 4y)^7?

7 3 * (-3)^3 * (4)^4 = -241920 combinatorial identity * first term to the power of n-k * second term to the power of k

Suppose r is a rational number. The product of any two rational numbers is rational. Therefore r^2 = r · r is also rational.

"Assuming facts that have not yet been proven"

If n is an odd integer, then n = 2k+1 for some integer k. Let n^2 = 2j + 1 for some integer j. Since n^2 is equal to two times an integer plus 1, then n^2 is odd.

"Circular reasoning"

m = 8 is an even integer since 8 = 2 · 4. m^2 = 8^2 = 64 is an even integer since 64 = 2 · 32. Therefore if n is an even integer, then n2 is also an even integer.

"Generalizing from examples"

If n is an odd integer, then n = 2k+1 for some integer k. Therefore n^2 = (2k+1)^2 and n^2 is odd.

"Skipping steps"

/xy/ =

/x + /y (DeMorgan's Law)

0 * 1

0 (multiplication acts as AND)

-11 mod 4

1

−344 mod 5

1

0 + 1

1 (addition acts as OR)

What is the value of 6 + 8 in Z4?

2

The 6th row of Pascal's Triangle is: 1 6 15 20 15 6 1 What is 7 4 ?

35

17 mod 6

5

What is the value of 5 times 7 in Z9?

8

a product of sums of literals ex: (x + y)(z + x)

CNF

a sum of products of literals ex: xy + zx

DNF

In an inductive proof of the theorem, what must be proven in the inductive step?

For all positive integers k, Q(k) implies Q(k + 1)

In an inductive proof of the theorem, what must be proven in the base case?

Q(1) is true

A license plate has 7 characters. Each character can be a capital letter or a digit except for 0. How many license plates are there in which no character appears more than once and the first character is a digit? (a) 9 · P(35, 6) (b) 9 · P(34, 6) (c) 9 · (35)^6 ) (d) 9 · (34)^6 )

a

A state's license plate has 7 characters. Each character can be a capital letter (A-Z), or a digit except for 0 (1-9). How many license plates are there in which exactly 3 of the 7 characters are digits? (a) 7 3 · (35)^4 (b) P(7, 3) · (35)^4 (c) 7 3 · (26)^4 (d) 7 3 · 9^3 · (26)^4

a

Because both {addition, complement} and {multiplication, complement} are functionally complete, a Boolean expression in DNF or CNF can be converted to CNF or DNF, respectively. (a) True (b) False

a

Literals grouped under a bar count as a CNF expression. (a) True (b) False

a

Provided set A1 = {1, 3, 2, 5} and A2 = {2, 5, 4, 6}, A3 = {2, 3, 5, 7}, then ∪ 3 i=1Ai = (a) {1, 2, 3, 4, 5, 6, 7} (b) {1, 3, 2, 5, 2, 5, 4, 6, 2, 3, 5, 7} (c) {2, 5} (d) {{1, 3, 4, 6}, {4, 5, 6, 7}, {1, 7}}

a

The law that establishes the set equality A ∩ /(B ∪ C)/ = A ∩ (/B ∩ /C) is (a) De Morgan's law (b) Associative law (c) Absorption law (d) Distributive law

a

The logic expression p → q means that p cannot be True when q is False. (a) True (b) False

a

Which of the followings does NOT have the same meaning of p → q? (a) q if p (b) p is sufficient for q (c) p is necessary for q (d) ¬q → ¬p

a

Which of the followings is a well-defined algebraic function from R to R? (a) f(x) = √ x^2 (b) f(x) = √ x (c) f(x) = 1 / x^2 − 2 (d) None of the above

a

Select the function that is NOT Ω(n^2 ) (a) 2^n (b) n + 17 log n (c) 6n log n + 3n^2 + 2 (d) 2n + n!

a (check this one)

What is a proposition?

a statement that is either true or false

OR gate

adds literals

A compound proposition with 5 propositional variables. The number of rows in its truth table is (a) 2 × 5 = 10 (b) 2^5 = 32 (c) 5^2 = 25 (d) None of the above

b

A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. Each party must elect a chair and a vice chair from their party's members, and one person cannot be elected for both. How many different outcomes are there for the chair and vice chair elections? (a) 100 · 99 · 98 · 97 (b) 44 · 43 · 56 · 55 (c) 100^4 (d) 44^2 · 56^2

b

Provided x = 3^4 · 5^3 · 7 · 11^4 and y = 3^2 · 5^4 · 7 · 13^4 . What is the prime factorization for gcd(x, y)? (a) 3^2 · 5^3 (b) 3^2 · 5^3 · 7 (c) 3^2 · 5^3 · 7^2 (d) 3^2 · 5^3 · 7 · 11^4 · 13^4

b

Select the converse of p → q (a) p → q (b) q → p (c) ¬p → ¬q (d) ¬q → ¬p

b

There is a set of 10 jobs in the printer queue. Two of the jobs in the queue are called job A and job B. How many ways are there for the jobs to be ordered in the queue so that job A completes some time before job B? (a) 10!/2 (b) 2 · 9! (c) 9! (d) 10!

b

A function is ___ if it is both one-to-one and onto

bijective

the coefficient of a^k(b^n-k) in (a+b)^n is n k

binomial theorem

Identify which line has a mistake in the proof of the theorem: The difference between two odd numbers is even. 1. Let x and y be two odd integers. We shall show that x − y is even. 2. Since x is odd, then x = 2k + 1 for some integer k. Since y is odd, then y = 2j + 1 for some integer j. 3. Let x − y = (2k + 1) − (2j + 1). 4. Since x−y is two times an integer, then x−y is even. (a) Line 1 (b) Line 2 (c) Line 3 (d) Line 4

c

Select the description that fits the sequence below: 8, 5, 2, 2, 1, -1 (a) Non-decreasing but not increasing (b) Non-increasing and decreasing (c) Non-increasing but not decreasing (d) Non-decreasing and increasing

c

What is the value of ((131)^39 + 11 · (−11)) mod 13? (a) 3 (b) 9 (c) 10 (d) 23

c

f is a function that maps 6-bit binary strings to 4-bit binary strings. For x ∈ {0, 1}^6 , f(x) is the string x with the last two bits removed. Which property best describes the function f? (a) Bijection (b) 2-to-1 correspondence (c) 4-to-1 correspondence (d) 6-to-1 correspondence

c

Provided f(n) = 100n+ (log n)^2 and g(n) = n + log 10n. Then, which of the followings best describes f and g? (a) f = Θ(g) (b) f = Ω(g) (c) f = O(g) (d) None of the above

c (check this one)

Select the function that is O(2^n) (a) 3^n (b) n 5 + 2n log n (c) n! (d) 2^n log n + n

c (check this one)

Select the function that is Θ(n log n) (a) 23n log log n + 3n log n (b) 15n + 17 log n (c) 6n log n + n^1.1 + 2 (d) 2^n log n + n

c (check this one)

n k = n (n-k)

combinatorial identity example

x is congruent to y mod m if x mod m = y mod m

congruence relation

A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. How many ways are there to pick a 10 member committee of senators with the same number of Reds as Blues? (a) 44 10 · 56 10 (b) 100 10 (c) 44 5 + 56 5 (d) 44 5 · 56 5

d

Let f : A → X be a function, then, which of the followings is not correct? (a) If |A| = |X|, then f must be a bijection. (b) If |A| > |X|, then f cannot be one-to-one. (c) If |A| < |X|, then f cannot be onto. (d) If |A| < |X|, then f may or may not be one to one.

d

Let f : A → X be a function, where A = {a, b, c, d}, and X = {1, 2, 3, 4}. Then, which of the followings about the function f = {(a, 3),(b, 2),(c, 4),(d, 3)} is correct? (a) It is one-to-one. (b) It is an onto function. (c) It is a bijection. (d) None of the above.

d

Natasha is in a class of 30 students that selects 4 leaders. How many ways are there to select the 4 leaders so that Natasha is one of the leaders? (a) 30 4 (b) 29 4 (c) 30 3 (d) 29 3

d

One mixed his steps in proving n! > 2^n for n ≥ 4 with mathematical induction. The right order of his steps numbered below should be: 1) Assume k! > 2^k for k ≥ 4. 2) Therefore, for any n ≥ 4, n! ≥ 2^n . 3) When n = k+1, n! = (k+1)! = k!(k+1) > 2^k (k + 1) > 2^k+1. The last step is because of k + 1 > 2. 4) When n = 4, n! = 4! = 24 and 2^n = 24 = 16. Hence, n! > 2^n for n = 4. (a) 3, 4, 2, 1 (b) 4, 3, 1, 2 (c) 2, 1, 3, 4 (d) 4, 1, 3, 2

d

Which number is congruent to 5639 mod 13? (a) 5621 (b) 5627 (c) 5653 (d) 5652

d

Which of the followings is correct? (a) log5 k + log5 2 = log5 (k + 2) (b) log5 k − log5 2 = log5 (k − 2) (c) (log5 k)/(log5 2) = log5 (k/2) (d) log5 k 2 = 2 log5 k

d

Which of the followings is not correct? (a) x + yz = (x + y)(x + z) (b) x(y + z) = xy + xz (c) (x + y)(x + y) = x (d) x + y + xy = x + y

d

Which statement is the contrapositive of: "If x = 4, then 3x = 12." (a) If x = 4 then 3x = 12. (b) If 3x = 12 then x = 4. (c) If x /= 4 then 3x /= 12. (d) If 3x /= 12 then x /= 4

d

the hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption

direct proof

intersection is empty (A ∩ B = ∅)

disjoint

floor ALWAYS rounds

down

7 | 50

false

An expression in DNF is always in CNF.

false

the set of all elements that are elements of both A and B; ∩

intersection

Suppose there is a k-to-1 correspondence from a finite set A to a finite set B. Then |B| = |A|/k.

k-to-1 rule

a way of ordering n-tuples in which two n-tuples are compared according to the first entry where they differ, goes smallest to largest ex: the word "comment" appears before the word "compare" because the first place where the two words differ is the 4th character and "m" appears before "p" in the alphabet

lexicographic order

AND gate

multiplies literals

to find r-subsets from set S ("n choose r"):

n! / r!(n-r)!

N

natural numbers; all numbers greater than or equal to 0

inverter gate

negates literals

Let X = {x, y, z}. Is the string zzyzx an element in X^4?

no

If f maps an element of the domain to zero elements or more than one element of the target

not well defined

A function f: X → Y is ___ or ___ if x1 ≠ x2 implies that f(x1) ≠ f(x2)

one-to-one (every Y has only one mapping)

if the range of f is equal to the target Y

onto (every Y is mapped to an X)

a non-empty set A is a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets

partition

n k = (n-1) (k-1) + (n-1) k

pascal's identity

If a function f has a domain of size at least n+1 and a target of size at most n, where n is a positive integer, then there are two elements in the domain that map to the same element in the target (i.e., the function is not one-to-one)

pigeonhole principle

Let A1, A2,...,An be finite sets. Then, |A1 × A2 × ... × An| = |A1| · |A2| · ... · |An|

product rule (multiple choices but multiple selection combinations)

proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true

proof by contrapositive

If A ⊆ B and there is an element of B that is not an element of A

proper subset

n div d produces the:

quotient

a sequence of r items with no repetitions, all taken from the same set

r-permutation

another word for subset, usually for counting situations in which there in no particular ORDER imposed on a set of outcomes

r-subset

Q

rational numbers; all real numbers that can be expressed as a/b

R

real numbers

n mod d produces the:

remainder

to write in sum-of-minterms form:

select the range values that result to 1

Z

set of all integers; ...-2, -1, 0, 1, 2...

If every element in A is also an element of B

subset

Consider n sets, A1, A2,...,An. If the sets are mutually disjoint (Ai ∩ Aj = ∅ for i ≠ j), then |A1 ∪ A2 ∪ ... ∪ An| = |A1| + |A2| + ... + |An|

sum rule (multiple choices but one selection)

the set of elements that are a member of exactly one of A and B, but not both; ⊕

symmetric difference

normal parenthesis indicate that:

the order of the elements does matter (usually expanded)

curly brackets indicate that:

the order of the elements does not matter (non-expanded permutations)

8 | 40

true

8 ∤ 79

true

An expression in CNF is always in DNF.

true

the set of all elements that are elements of A or B; ∪

union

ceiling ALWAYS rounds

up