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Cis2910
| Question | Answer |
|---|---|
| Neil James Alexander Sloane 1939 - idk | Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences (OEIS). his doctoral dissertation was titled Lengths of Cycle Times in Random Neural Networks |
| Fibonacci 1170 to 1240 | he was the most talented Western math-matician of the Middle Ages, fibbonacci is a nickname from filius Bonacci son of Bonacci he composed in 1202 of Liber Abaci Book of Calculation and he introduced Europe to Fibonacci numbers |
| Blaise Pascal (1623–1662 | one of the first two inventors of the calculator and is best known for pascals triangle (each number is the sum of the two above it). |
| Sir Isaac Newton (1642-1726 | In addition to his work on calculus, as a mathematician, he contributed to the study of power series, and generalised the binomial theorem to non-integer exponents |
| Q = | p/q | p ∈ Z, q ∈ Z, q 6 = 0}, the set of rational numbers |
| What is russel's paradox | Is S an element of it's Self? This is the paradox of Consider the set of all sets that are not elements of themselves: S = {A | A is a set and A /∈ A} |
| what is the domain and what is the co-domain/target | domain left side of the function and the target is what is being mapped to |
| what conditions must be met for a function to be invertible? (Has an inverse ) | It must be a bijection or surjective and injective |
| f(n) = n! is known as what and how do we calculate it | factorial function and its equal to the product of the first n positive integers |
| An arithmetic progression is a sequence of the form: a, a+d, a+2d, a+3d, a+4d, . . . what are a and d | a and d are the common differences |
| A geometric progression is a sequence of the form: a, ar, ar^2, a^r3, ar^4, . . what are a and r | a and r are the common ratio |
| what is a recurrence relation? | A recurrence relation for the sequence {an} is an equation that expresses an in terms of one or more of the previous terms |
| what is the recurrence relation for.. Sequence of prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, . . . The number of possible chess games after n moves. 20, 400, 8902, 197742, 4897256, 120921506, . . . | trick question, there is no known recurrence for these relations |
| What is this sequence? 1, 2, 3, 4, 6, 8, 11, 13, 16, 18 | Ulam sequence, for n>2 each term is the sum of two distinct earlier terms |
| What is this sequence? 1, 4, 11, 16, 24, 29, 33, 35, 39, 45, 47, 51, 56, . . | Aronson’s sequence, t is the first, fourth, eleventh, sixteenth, ... letter of this sentence |
| what is this sequence? 6, 28, 496, 8128, 33550336, 8589869056 | Perfect numbers, numbers that are equal to the sum of every smaller number that divides them |
| What are the sum operations? | 1. Splitting a sum (a+b) -> sum(a) + sum (b) 2. Pulling out multiplicative constants (c* ai) -> c* sum ai 3. Changing the index of summation -> replace i with i +1 |
| for Arithmetic series ∑ i=1 i = 1 + 2 + 3 + · · · + n what is the formula | n(n + 1)/2 |
| for sum of square (sum with i^2) what is the formula | n(n + 1)(2n + 1)/6 |
| for sums involving r^i what formula is used | r^n+1 − 1 / r − 1 |
| theres a special case for sums involving 2^i what is this formula | 2^n+1 -1 |
| What is the simple definition of induction | reasoning from the particular to the general, what's true for particulars (base cases should be true for more general) note that it only involves verifying conjectures based on positive integers |
| For proofs by induction To prove P (n) is true for all n ≥ 1, we must prove: what is P(k ) called? | 1. Basis step: P (1) is true and 2. Inductive step: for all k ≥ 1, P (k) ⇒ P (k + 1) P(k) is called the inductive hypothesis |
| What is the well ordering property? | Every nonempty set of non-negative integers has a least element or every set of 0+ that is non empty has at least one element this is applied to round robin tournaments |
| What is combinatorics the study of? Enumeration is an important area what is this the study of? | arrangement of objects the counting of objects with certain properties |
| What is the product rule with counting | # of ways to do a task = # of ways to do first task * ways to do second after the first |
| What is the sum rule with counting | if a task can be either done in one way or another way then add th total ways together |
| What is the pigeon hole principal | If N objects (pigeons) are placed into k boxes (holes), then there is at least one box containing at least ⌈N/k⌉ objects |
| What is a permutation? | A permutation of a set of distinct objects is an ordered arrangement of these objects |
| All permutations of {1, 2, 3} | 123, 132, 213, 231, 312, 321 |
| What is an r- permutation | An ORDERED arrangement of r elements of an n-set is called an r-permutation |
| r-permutations formula P(n, r): when should it be used | P (n, r) = n! / (n - r)! when the order is relevant and the objects arent all the same |
| What is an r combination | An UNORDERED arrangement of r elements |
| r-combination formula C(n, r) or n choose r: when should it be used | C(n, r) = n! / r!(n-r)! when choosing a number of the same objects |
| Lexico ordering | the standard smallest to largest ordering by the value of the strings |
| What is the formula for ways to choose if repetition is allowed (ties ) | (n + r - 1) choose r or (n + r -1 ) choose n - 1 |
| The sequence starting 0, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, … corresponds to: | the number of series-reduced trees with n nodes |
| The sum 2^10 + 2^11 + 2^12 + …. + 2^20 can be simplified to: a) 2^21-1 b) 2^21 - 2^10 c) 2^20 + 2^10 - 1 d) 2^20 - 2^10 | b) 2^21 - 2^10 |
| The sequence starting 2,2,4,6,10,16,26,42, … can best be described as: | a sequence similar to the Fibonacci sequence |
| The sequence starting 6,18,54,162, ... can best be described as: arithmetic or geometric | a geometric progression |
| The sum of the EVEN integers in {1,2, … ,99,100} is | between 2500 and 2750 Equals 2 times the sum of the first 50 integers = 2 * (50)(51)/2 = 2550 |
| The number of binary strings of length n with no 00 substrings is the same as the n-th Fibonacci number. T or F | False |
| How many binary strings of length 5 begin with 00 or end with a 1? a) 32 b) 24 c) 20 d) 16 | Inclusion exclusion. 2^3 + 2^4 - 2^2 = 8 + 16 - 4 = 20 |
| What was William Rowan known for | (1805-1865) His work is fundamental to modern theoretical physics, particularly his form Newtonian mechanics. Hamiltonian paths and cycles are named after him he invented the game, now also known as Hamilton’s puzzle |
| What was Leonhard Euler known for | Euler is credited with being the first to develop graph theory from a solution for the problem of the Seven Bridges of K ̈onigsberg, which is also considered the first practical application of topology, he also discovered the formula V − E + F = 2 r |
| What was Jacob (James) Bernoulli known for | his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi. The term Bernoulli trial resulted from this work. |
| T or F each graph has a unique spanning true | False graphs ca have multiple spanning trees they are not unqiue |
| what does the peterson graph look like | a k5 with and a petagon cycle around the outside |
| What is the binomial formula and when should it be used | P(X=k)=C(n, k)p^k(1−p)^(n−k) n = fixed number of trials like 10 trials k = # of successes or what your trying p = probability of what your trying to measure |
| What formula would you use for 10 flipps with equal probability to get 4 heads | Binomial formula |
| What is the chromatic number of a graph | the least colouring, |
| How do you find the chromatic number | 3 cycle/ triangle -> at least 3 bipartite -> 2 most common just test it and see if you can get a least colouring |
| how many edges are in a spanning tree | n -1 |
| What was prim most known for | Known for Prim’s Algorithm It finds a minimum spanning tree (MST) in a weighted graph |
| What was Kruskal known for | Known for Kruskal’s Algorithm Also finds a minimum spanning tree It works by Sorting all edges by weight and Adding the smallest edges as long as they don’t form a cycle |
| What is a hamiltonian cycle | cycle that touches every vertex exactly once there is no simple rules/ conditions |
| What is a Euler cycle | touches every edge exactly once |
| What are the conditions of a euler cycle | every vertex must have even degree\ must be connected |
| What are the conditions of a euler trail | graph is connected has exactly 0 or 2 odd degree vertices |
| is every euler cycle a euler trail | yes |
| what is a planar graph and what are the conditions for it to be planar | a graph with no crossing edges euler formula must hold E <= 3V - 6 |
| What are some specila case non planar graphs | K5 and K3,3 if a graph contains these its immediately not planar |
| What are the conditions for bipartite graphs | if you find an odd length cycle like 3 not bipartite |
| what is the most commonway to find the probability of a specific outcome | event space / sample space |
Created by:
jayne_