173 final study

Justin needs to pick 17 toy animals to give to children at a party. The animals come in 5 kinds: dogs, dinosaurs, cows, lizards, and fish. How many different ways can he choose his set of toys?

(17+4 4)

The number of ways to select a set of 17 flowers chosen from 4 possible varieties (zero or more of each variety). (17 5)/(20 4)/(20 3)/(17 4)/(21 4)/17!/4!

(20 3)

Margaret's home is defended from zombies by wallnuts, peashooters, and starfruit. At each timestep, she can make one move, which adds or deletes one plant from her arsenal. If she starts with 7 wallnuts, 2 peashooters, and 10 starfruit, how many different sequences of 20 moves will get her to a configuration with 4 wallnuts, 10 peashooters, and 19 starfruit?

(20 3)(17 8)

The level of the root node in a tree of height h. 0/1/h − 1 h h/h+1

0

n−1Σk=1 1/2^k = 1 − (1/2)^n | 2 − (1/2)^n | 1 − (1/2)^n−1 |2 − (1/2)^n−1

1 − (1/2)^n−1

The chromatic number of the 3-dimensional hypercube Q3 1/2/3/4

2

The running time of mergesort is recursively defined by T(1) = d and T(n) = 2T(n − 1) + c/2T(n − 1) + cn/2T(n/2) + c/2T(n/2) + cn

2T(n/2) + cn

sum of 2^k, from k = 0 to n-1 = 2^n − 2/2^n − 1/2^n−1 − 1/2^n+1 − 1

2^n - 1 (remember: sum of x^k = x^k+1 - 1/x-1)

n+1Σk=1 2^k = 2^n+1 + 1/2^n+2 − 1/2^n+2 − 2/2^n − 2

2^n+2 - 2

n+1Σk=1 2^k = 2^n+1 + 1/2^n+2 − 1/2^n+2 − 2/2 n − 2

2^n+2 − 2 (notice you have to subtract the 0th term because it starts at 1 instead)

The diameter of a full, complete 7-ary tree of height h. ≤ h/h/h + 1/2h/7h/7h + 1

2h

The diameter of a full, complete tree of height h. ≤ h/h/h + 1/2h/≤ 2h

2h

The number of edges in the 4-dimensional hypercube Q4 5/12/32/64

32 n2^n-1 (4*2^3)

The running time of Karatsuba's algorithm is recursively defined by T(1) = d and T(n) = 2T(n/2) + cn/3T(n/2) + cn/4T(n/2) + cn/4T(n/2) + c

3T(n/2) + cn

The real numbers are countable true/false/not known

FALSEEEEEEE PERDIODTTTTTT

All elements of M are also elements of X. M = X/M ⊆ X/X ⊆ M

M ⊆ X

3 n is Θ(5^n)/O(5^n)/neither of these

O(5^n)

If f : R → P(Z) then f(17) is an integer/a set of integers/one or more integers/a power set

a set of integers

∅ is an element of Z/a subset of Z/both/neither

a subset of Z

A full m-ary tree with i internal nodes has mi + 1 nodes total. always/sometimes/never

always

If xRy is never true, then the relation R is symmetric/antisymmetric/both/neither

both

Q × {π, √ 2} finite/countably infinite/uncountable

countably infinite

The set of (unlabelled, finite) binary trees with exactly 4 leaves. finite/countably infinite/uncountable

countably infinite

The set of all (finite) phone lattices using the 26 letters A, ..., Z finite/countably infinite/uncountable

countably infinite

The set of all (finite, unlabelled) graphs, where isomorphic graphs are treated as the same object. finite/countably infinite/uncountable

countably infinite

The set of all finite lists of integers. finite/countably infinite/uncountable

countably infinite

The set of all finite sequences of Chinese characters. finite/countably infinite/uncountable

countably infinite

The set of all finite-length strings of decimal digits finite/countably infinite/uncountable

countably infinite

The set of all full binary trees where each node contains one of the letters A, B, or C is finite/countably infinite/uncountable

countably infinite

The set of all polynomials with rational coefficients. finite/countably infinite/uncountable

countably infinite

A piano tune is a finite sequence of notes found on the standard piano keyboard. The set of all piano tunes is: finite/countably infinite/uncountable

countably infinite (like the set of finite length strings made from a finite alphabet)

The set Q^2 is finite/countably infinite/uncountable?

countably infinite (the rationals :DD)

Any function from N to {0, 1} has a corresponding C++ program that computes it. true/false/not known

false

Every function from the integers to the integers has a corresponding finite-length formula true/false/not known

false

Every function from {1, 2, 3} to the reals has a finite formula. true/false/not known

false

Every real number has a corresponding finite formula. true/false/not known

false

If a function from R to R is increasing, it must be one-to-one. true/false

false

Suppose a graph with 12 vertices is colored with exactly 5 colors. By the pigeonhole principle, every color appears on at least two vertices. true/false

false

The rational numbers have the same cardinality as the reals. true/false/not known

false

Two positive integers p and q are relatively prime if and only if gcd(p, q) > 1. true/false

false

|N^2| < |N ^3| true/false/not known

false

For any integers p and q, if p | q then p ≤ q. true/false

false (0 is an integer that can't be p)

The set of board configurations for the game of chess finite/countably infinite/uncountable

finite

The set of chords (simultaneous combinations of notes) playable on an 88-key piano. finite/countably infinite/uncountable

finite

The set of netIDs currently in use at U. Illinois. finite/countably infinite/uncountable

finite

If f : Z → R is a function such that f(x) = 2x then the set of all even integers is the of f. domain/co-domain/image

image

If xRx is never true, then the relation R is reflexive/irreflexive/both/neither

irreflexive

Suppose I want to estimate 103/20. 3 is an upper bound/lower bound/exact answer/not a bound

lower bound (3 x 20 is lower than 103)

n! O(2^n)/Θ(2^n)/neither of these

neither of these

Wn has a Euler circuit. always/sometimes/never

never

If A is countably infinite, then P(A) is countably infinite never/sometimes/always

neverrrr

f : N → N, f(x) = 3 − x one-to-one/not one-to-one/not a function

not a function (not all the inputs have an output)

f : R → Z, f(x) = x one-to-one/not one-to-one/not a function

not a function (some inputs won't have an output, not good)

Problems in NP need exponential time proven true/proven false/not know

not known

The Marker Making problem can be solved in polynomial time. true/false/not known

not known (it's np)

g : Z → Z, g(x) = |x| one-to-one/not one-to-one/not a function

not one-to-one

f : N → R, f(x) = x^2 + 2 onto/onto/not a function

not onto (onto means if the function's image is its entire co-domain)

g : Z → Z, g(x) = 7 − x/3(floor function) onto/not onto/not a function

onto

n log base 2 of 3 grows: faster than n^2/slower than n^2/at the same rate as n^2

slower than

Cn is bipartite always/sometimes/never

sometimes

If A is countable, then P(A) is countable. always/sometimes/never

sometimes

7 | 0 true/false

true

7 ≡ 5 (mod 1) true/false

true

For any positive integers p, q, and k, if p ≡ q (mod k), then p 2 ≡ q 2 (mod k) true/false

true

For any real number x, if x > 10, then x^2 > 0.

true

If √ 2 is rational, then −3 is positive. true/false/undefined

true

N 2 has the same cardinality as N. true/false/not known

true

P(Q) has the same cardinality as the reals. true/false/not known

true

Suppose A is a non-empty set. Then P(A) is larger than A. true/false/true for finite sets

true

There exist mathematical functions that cannot be computed by any C program. true/false/not known

true

f : A → B is one-to-one if and only if |A| ≤ |B|. true/false/true for finite sets

true

p → q ≡ ¬q → ¬p true/false

true

|A × A| ≥ |A| true/false/true for some sets

true

∀x ∈ R, if π = 3, then x < 20. true/false/undefined

true

There are mathematical functions that don't have a finite formula. true/false/not known

true (formula is just a

|A × A| > |A| true/false/true for some sets

true for some sets

|A ∪ B| = |A| + |B| true for all sets A/true for some sets A/false for all sets A

true for some sets A

The running time of mergesort is O(n^3) true/false

true Θ(n log n) which is < O(n^3)

The rational numbers have the same cardinality as the integers. true/false/not known

true!!!

All infinite sequences of emojis. finite/countably infinite/uncountable

uncountable

P(N) finite/countably infinite/uncountable

uncountable

The interval [2, 3] of the real line. finite/countably infinite/uncountable

uncountable

The irrational numbers finite/countably infinite/uncountable

uncountable

The real numbers finite/countably infinite/uncountable

uncountable

The set containing all functions f from the set of even integers to the set of even integers is finite/countably infinite/uncountable

uncountable

The set of all polynomials with real coefficients. finite/countably infinite/uncountable

uncountable

The set of all intervals [a, b] of the real line finite/countably infinite/uncountable

uncountable (IT"S REALLS NOS)

The complex numbers finite/countably infinite/uncountable

uncountable (a subset of the reals i think)

Putting 10 people in the canoe caused it to sink. 10 is _____ how many people the canoe can carry. an upper bound on/exactly/a lower bound on/not a bound on

upper bound

If f : A → B is one-to one: |A| ≤ |B|/|A| ≥ |B|/|A| = |B|

|A| ≤ |B|

If f : A → B is one-to one |A| < |B|, |A| ≤ |B|, |A| = |B|

|A| ≤ |B| (one-to-one means it never assigns two input values to the same output value. Or, said another way, no output value has more than one pre-image)

T(1) = d T(n) = 2T(n − 1) + c Θ(n) Θ(n^2 ) Θ(n log n) Θ(2^n)

Θ(2^n) (towers of hanoi)

Karatsuba's integer multiplication algorithm Θ(n^2) Θ(n^3) Θ(n log n) Θ(n^log base 2 of 3 ) Θ(n^log base 3 of 2 ) Θ(2^n)

Θ(n^log base 2 of 3)

Σk=3 to n of k^7 = Σp=1 to n-2 of p^9/Σp=1 to n-2 of k^7/Σp=1 to n-2 of k^9/Σp=1 to n-2 of (p+2)^7

Σp=1 to n-2 of (p+2)^7 (variable has to be p, obviously not p^9)

Chromatic number of G = C(G)/φ(G)/χ(G)/||G||

χ(G) KEYBLADE

{13, 14, 15} × ∅ = ∅/{∅}/{13, 14, 15}

∅

Chromatic number of Cn. 2/3/≤ 3/≤ 4

≤ 3

The chromatic number of a graph with maximum vertex degree D = D/ = D + 1/≤ D + 1/≥ D + 1

≤ D + 1

Total number of leaves in a 3-ary tree of height h 3^h/≤3^h/ 1/2(3h+1 − 1)/3^h+1 − 1

≤3^h