discrete 8.1-8.6 and 8.8-8.9

2 components of an inductive proof

1) base case, and 2) inductive step

3 components of a recursive definition of a set

1) basis 2) recursive rule 3) exclusion statement

basis (component of a recursive definition of a set)

A basis explicitly states that one or more specific elements are in the set.

explicit formula

A sequence can be specified by an explicit formula showing how the value of term ak depends on k.

arithmetic sequence

An arithmetic sequence is a sequence of real numbers where each term after the initial term is found by taking the previous term and adding a fixed number called the common difference. An arithmetic sequence can be finite or infinite.

exclusion statement (component of a recursive definition of a set)

An exclusion statement states that an element is in the set only if it is given in the basis or can be constructed by applying the recursive rules repeatedly to elements given in the basis.

Closed form for the sum of terms in a geometric sequence

For any real number r ≠ 1 and any integer n ≥ 1, the summation of the geometric sequence a*r^k from (k = 0) to (n-1) can be calculated: a(r^(n) - 1)/(r-1)

recursive definition

In a recursive definition of a function, the value of the function is defined in terms of the output value of the function on smaller input values.

inductive hypothesis

In the statement "S(k) implies S(k+1)" of the inductive step, the supposition that S(k) is true is called the inductive hypothesis.

principle of mathematical induction

Let S(n) be a statement parameterized by a positive integer n. Then S(n) is true for all positive integers n, if: 1) S(1) is true (the base case). 2) For all k ∈ Z+, S(k) implies S(k+1) (the inductive step).

Thm: number of vertices in a perfect binary tree

Let T be a perfect binary tree. Then the number of vertices in T is 2^(k) - 1 for some positive integer k.

Recursion

Recursion is the process of computing the value of a function using the result of the function on smaller input values.

structural induction

Structural induction is a type of induction used to prove theorems about recursively defined sets that follows the structure of the recursive definition.

base case

The base case establishes that the theorem is true for the first value in the sequence.

base case (for strong induction)

The base case for a proof by strong induction establishes that S(n) holds for n = a through b, where a and b are constants.

inductive step

The inductive step establishes that if the theorem is true for k, then the theorem also holds for k + 1.

inductive step (strong induction)

The inductive step in a proof by strong induction assumes that S(j) is true for all values of j in the range from a through some integer k ≥ b and then proves that theorem holds for k+1.

closed form

A closed form for a sum is a mathematical expression that expresses the value of the sum without summation notation.

Are geometric sequences finite or infinite?

A geometric sequence can be either finite or infinite.

geometric sequence

A geometric sequence is a sequence of real numbers where each term after the initial term is found by taking the previous term and multiplying by a fixed number called the common ratio.

recursive rule (component of a recursive definition of a set)

A recursive rule shows how to construct larger elements in the set from elements already known to be in the set. (There is often more than one recursive rule).

recurrence relation

A rule that defines a term an as a function of previous terms in the sequence is called a recurrence relation.

sequence

A sequence is a special type of function in which the domain is a consecutive set of integers.

decreasing

A sequence is decreasing if for every two consecutive indices, k and k + 1, in the domain, ak > ak+1.

increasing

A sequence is increasing if for every two consecutive indices, k and k + 1, in the domain, ak < ak+1.

non-decreasing

A sequence is non-decreasing if for every two consecutive indices, k and k + 1, in the domain, ak ≤ ak+1.

non-increasing

A sequence is non-increasing if for every two consecutive indices, k and k + 1, in the domain, ak ≥ ak+1.

finite sequence

A sequence with a finite domain is called a finite sequence.

infinite sequence

A sequence with an infinite domain is called an infinite sequence.

quotient and remainder theorem

Let n be an integer and d a positive integer. There are integers q and r such that n = qd + r and 0 ≤ r < d.

Closed form for the sum of terms in an arithmetic sequence

For integer n ≥ 1, the summation of the arithmetic sequence (a + kd) from (k = 0) to (n-1) can be calculated: an + [d(n-1)n/2]

principle of mathematical induction (informally defined)

The principle of mathematical induction states that if the base case (for n = 1) is true and inductive step is true, then the theorem holds for all positive integers.

principle of strong induction

The principle of strong induction assumes that the fact to be proven holds for all values less than or equal to k and proves that the fact holds for k+1. By contrast the standard form of induction only assumes that the fact holds for k in proving that it holds for k+1.

well-ordering principle

The well-ordering principle says that any non-empty subset of the non-negative integers has a smallest element.