Discrete Math Chapter 4: Functions.

Identity Function

A function that always maps a set onto itself and maps every element onto itself.

Range of f

Possible Y outcomes.

Onto (Surjective) Function

The range of f is equal to the target Y. That is, for every y ∈ Y, there is an x ∈ X such that f(x) = y.

Domain of f (if f is a function that maps a set X to a set Y)

The set X.

Target of f (if f is a function that maps a set X to a set Y)

The set Y.

Strictly Increasing (Function)

Whenever x1 < x2, then f(x1) < f(x2). If b > 1, then the functions f(x) = bx and f(x) = logb (x) are both strictly increasing. The fact that both functions are strictly increasing can help in approximating the value of the functions. For example, the value of log3200 is not an integer and would be difficult to determine without a calculator. However, since 3^4 = 81 and 3^5 = 243, by definition log3 (81) = 4 and log3 (243) = 5. Because the log function is strictly increasing and 81 < 200 < 243, the value of log3 (200) is in between 4 and 5.

Properties of Exponents

b^x(b^y) = b^(x+y) (b^x)^y = b^(xy) (b^x/b^y) = b^(x−y) (bc)^x = b^x(c^x)

Divide-and-Conquer

A common strategy in computer science in which a problem is solved for a large set of items by dividing the set of items into two evenly sized groups, solving the problem on each half and then combining the solutions for the two halves. For example, one approach to sorting a list of numbers is to divide the list in half, sort each half separately and then merge the two sorted lists. Note: The logarithm function plays an important role in analyzing divide-and-conquer algorithms.

Bijective Function (One-to-One Correspondence)

A function that is both one-to-one and onto. A bijective function is called a bijection. A bijection is also sometimes called a one-to-one correspondence. Notice that if a function is a bijection then the size of the domain is the same as the size of the target: If function f: A → B is a bijection and A is finite, then |A|=|B|.

Arrow Diagram

An arrow diagram, the elements of the domain X are listed on the left and the elements of the target Y are listed on the right. There is an arrow from x ∈ X to y ∈ Y if and only if (x, y) ∈ f.

Properties of Logarithms

Logb (xy) = Logb (x) + Logb (y) Logb (x/y) = Logb (x) - Logb (y) Logb (xy) = y Logb (x) Logc (x) = (Logb (x))/(Logb (c))

g: R → R such that g(x) = |x|.

Note that g maps every real number to a real number. However, g does not map any number to a negative number.

Inverse (of a Function)

Obtained by exchanging the first and second entries in each pair in f. The inverse of f is denoted by f^-1: f^-1 = { (b, a) : (a, b) ∈ f }. Note: Reversing each pair in f does not always result in a function. Therefore, some functions do not have an inverse. A function f: A → B has an inverse if and only if reversing each pair in f results in a function from B to A. The result of reversing each pair in f is a function if every element in B is mapped to exactly one element in A. A function f: A → B has an inverse if and only if f is a bijection. (Recall that a bijection is a function that is one-to-one and onto).

Exponent

The input x to the function b^x is called the exponent.

Logarithm Function

The inverse of the exponential function.

f: X --> Y

The notation to express the fact that f is a function from X to Y. The fact that f maps x to y (or (x, y) ∈ f) is denoted by f(x) = y.

Base of the Exponent

The parameter b is called the base of the exponent in the expression b^x.

Composition

The process of applying a function to the result of another function. Define functions f: X → Y and g: Y → Z. Then the composition of g with f, denoted g ο f, is the function (g ο f)(x): X → Z, such that (g ο f)(x) = g(f(x)).

Base of the Logarithm

b^x=y ⇔ logb y=x The parameter b is called the base of the logarithm in the expression logb y.

Ceiling Function

ceiling: R → Z. ceiling(x) = the smallest integer y such that y ≥ x. ceiling(x)=⌈x⌉ The ceiling function rounds a real number to the nearest integer in the upward direction.

Exponential Function

expb(x)=b^x Where b is a positive real number and b ≠ 1. Note: The exponential function is one-to-one and onto, and therefore has an inverse.

One-to-One (Injective) Function

f maps different elements in X to different elements in Y.

Floor Function

floor: R → Z. floor(x) = the largest integer y such that y ≤ x. floor(x)=⌊x⌋ The floor function maps a real number to the nearest integer in the downward direction.