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Algorithms and growth of functions
Big-O notation, comparing growth rates, and what "efficient" means.
Big-O compares functions by eventual growth: n² is O(n³), 2ⁿ dominates every polynomial, n! dominates 2ⁿ. Picture it: the graphs of n, n log n, n², 2ⁿ pulling apart. Think it: P vs NP asks whether a whole class of problems is polynomial, the Frontiers page.
Eżempju maħdum · limit of (2^n)/(n!) as n -> oo
Pass b'pass
- \lim_{n \to \infty^+-} \frac{2^{n}}{n!}
Try direct substitution first.
- \
As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- = 0
Take the limit.
Jiżvelaw it-tweġiba
Issa inti Agħżel problema, jew ittajpja jew iġbed tiegħek stess.Kull pass, stampa, it-tweġiba moħbija sakemm inti tistaqsi.
Kont b'xejn iżid noti fuq kull lezzjoni, rekord ta' dak li lest, problemi solvuti tiegħek f'post wieħed, u għalliem inti tista' tistaqsi dwar din il-paġna.Il-matematika nnifisha hija miftuħa għal kulħadd, log fil jew le.
Irreġistra issa LoginSimboli użati hawnhekk
Taptap kwalunkwe simbolu għad-definizzjoni sħiħa, stampa, u dak li kull ittra fiha tfisser.
Kif għandek tuża: Algorithms and growth of functions
- Try direct substitution first.
- As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- Take the limit.
Mistoqsijiet li n-nies jistaqsu
What makes mathematics "discrete"?
It deals with separate, countable objects (integers, graphs, statements) rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.
How does a proof by induction work?
Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.
What order should I take these lessons in?
Logic and proof first, because every later lesson proves things. Then induction, sets, relations and functions, counting, recurrences, graphs, Boolean algebra and automata, and finally algorithms and complexity, which use nearly everything before them.
Is discrete mathematics the same as the maths of computer science?
Largely. Data structures are graphs and trees, correctness arguments are induction, running times are recurrences and sums, circuits are Boolean algebra, and what a computer can do efficiently is the subject of the last lessons. It is also ordinary mathematics in its own right.
Do I need calculus for this course?
No. School algebra is enough. A little familiarity with limits helps when comparing growth rates, and the lesson on asymptotic analysis explains the one limit fact it uses.
Aktar fil Discrete Math & Logic
Truth tablesSums and inductionProof by inductionPredicate logic and quantifiersDirect proof, contrapositive and contradictionStrong induction and the well-ordering principleRelations: equivalence relations and partial ordersFunctions and cardinalityCounting: rules, choices and bijectionsThe pigeonhole principleSolving linear recurrence relationsGraphs: degrees, paths and connectivityTrees and spanning treesEuler and Hamilton paths