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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 |
| what does OEIS stand for and who created it | On-Line Encyclopedia of Integer Sequences, Neil James Alexander Sloane |
| Who created the 3 laws of motion, laws of gravity, established calculus, built the first reflective telescope and made the binomial theorem | Sir issac newton |
| set | is an unordered collection of objects |
| Z = | {. . . , −2, −1, 0, 1, 2, . . .}, the set of integers |
| Z+ = | {1, 2, . . .}, the set of positive integers |
| N = | {0, 1, 2, . . .}, the set of non-negative integers |
| Q = | p/q | p ∈ Z, q ∈ Z, q 6 = 0}, the set of rational numbers |
| R = | R, the set of real numbers |
| power set | s the set of all subsets of S. This set is denoted by P (S) |
| ordered n-tuple | (a1, a2, . . . , an) is the ordered collection that has a1 as its first element, a2 as its second element, . . ., an as its n-th element. |
| Let A = {0, 1, 2}, B = {a, b}, C = {1, 2} What is A × B ? | {(0, a), (0, b), (1, a), (1, b), (2, a), (2, b)} |
| Let A = {0, 1, 2}, B = {a, b}, C = {1, 2} What is B × A ? | {(a, 0), (a, 1), (a, 2), (b, 0), (b, 1), (b, 2)} |
| what is the principal of inclusion exclusion for the union of two sets |A ∪ B| | |A| + |B| − |A ∩ B| you must take away the intersections or else its counted twice |
| 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 |
| one to one or injective | everything in the domain is mapped to one thing in the target |
| surjective or onto | everything in the target has something mapped to it |
| 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 |
| How are sequences different than a set, what is a finite sequence called? | sequence is and ORDERED list of items and sets is an unordered listing, finite sequence is called a string |
| 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 |
| To find the first terms of a sequence what steps should be taken | 1. identify if its geometric or arthmetic (is it increasing very fast) |
| 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 |
| Inclusion/exclusion for 3 sets | |A1 ∪A2 ∪A3| = |A1|+|A2|+|A3|−|A1 ∩A2|−|A1 ∩A3|−|A2 ∩A3|+|A1 ∩A2 ∩A3| |
| 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): ORDERED | P (n, r) = n! / (n - r)! |
| What is an r combination | An UNORDERED arrangement of r elements |
| r-combination formula C(n, r) or n choose r: UNORDERED | C(n, r) = n! / r!(n-r)! |
| T or F 10 choose 2 = 10 choose 8 | True |
| What is gray code ordering | found by flipping one bit at a time |
| 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 |
| A proposition P(n) is true for all integers n >= 1 if (a) P(1) is true and (b) the implication P(k) -> P(k+1) evaluates to true for any k > 1. | False. This is almost the rule for induction, but we must have k>=1. Otherwise we do not know if P(2) is true. Recall the Horses example. |
| 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 |
Created by:
jayne_