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Proof by induction

Base case, inductive step, strong induction, and the well-ordering principle.

To prove P(n) for all n: prove P(1), then prove P(n) ⇒ P(n + 1). Strong induction assumes all earlier cases. Picture it: the staircase of dominoes. Think it: induction is equivalent to the well-ordering of ℕ. Every non-empty set of naturals has a least element.

Esempio svolto · sum of k for k = 1 to n

Calculus · Series-And-Sums

Sum k for k = 1 to n

\sum_{k=1}^{n} k

Passo dopo passo

  1. \sum_{k=1}^{n} k

    Write the sum out.

  2. 1 + 2 + 3 + 4 + \cdots

    The first few terms.

  3. = \frac{n \left(n + 1\right)}{2}

    Closed form (a standard summation formula, provable by induction).

Rivela la risposta
\frac{n \left(n + 1\right)}{2}

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Come: Proof by induction

  1. State P(n) precisely.
  2. Base case: verify P(1) (or P(0)).
  3. Inductive step: assume P(k), derive P(k + 1), citing the assumption exactly once at least.
  4. Conclude by the principle of induction.

Domande che la gente chiede

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.

Altro in Discrete Math & Logic

Truth tablesSums and inductionAlgorithms and growth of functionsPredicate 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