Functions and Sequences

¹ Functions

7.1 Definition and Notation

• Function: A function from a set A into a set B is a relation from A into B such that each element of A is related to exactly one element of the set B. The set A is called the domain of the function and the set B is called the codomain.
• The most important condition is: No element from the domain that have more than one image in the codomain, and all element of the domain must have images.
• Forms of function notations:
  • A Function as a list of ordered pairs:
    • Notation: f = {(a, b), (c, d), (e, f), ...} where a, c, e, ... are elements of the domain and b, d, f, ... are elements of the codomain.
    • Example: f = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} where f is a function that maps A = {1, 2, 3, 4, 5} to B = {2, 4, 6, 8, 10}.
  • A Function as a set of ordered pairs in set-builder notation:
    • Notation: f = {(x, y) | x ∈ A and y ∈ B} where A is the domain and B is the codomain.
    • Example: f = {(x, 3x) | x,y ∈ R} where f is a function that maps R to R where each element in R is mapped to 3 times itself.
  • The set of all possible functions from A to B:
    • Notation: B^A where B^A = {f | f is a function from A to B}.
    • Example: B^A = {f1, f2} where A = {1, 2, 3} and B = {a, b, c} and f1 = {(1, a), (2, b), (3, c)} and f2 = {(1, a), (2, c), (3, b)}.
  • A Function as a formula( A definition based on images):
    • Notation: f(x) = expression where x is an element of the domain and expression is a formula that defines the function the equals f = {(x, y) | x ∈ A and y ∈ B}
    • Example: f(x) = x^2 where f is a function that maps R to R where each element in R is mapped to its square.
• Range of a function:
  • The set of all images of the elements of the domain.
  • Notation: f(X) = {f(a) | a ∈ X} = {b ∈ Y | ∃a ∈ X such that f(a) = b} for function f: X → Y.
  • The range of a function is a subset of its codomain.
• Examples of Non functions relations:
  • Not all elements of the domain have images in the codomain:
    • Example: A={1, 2, 3} and B={a, b, c} and F={(1, a), (2, b)}.
    • F is not a function because 3 (of A) does not have an image in B.
  • An element of the domain has more than one image in the codomain:
    • Example: A={1, 2, 3} and B={a, b, c} and F={(1, a), (1, b), (3, b)}.
    • F is not a function because 1 (of A) has two images in B as (1, a) and (1, b).
    • The image of 1 is not well defined.
• Examples of Non functions:

7.2 Properties of Functions

• Injective Function, Injection:
  • A function f : A → B is injective if ∀a, b ∈ A, a ≠ b ⇒ f(a) ≠ f(b).
  • Also called one-to-one function.
  • The condition for injective function is: f(a) = f(b) ⇒ a = b.
  • That is, if two elements of the domain have the same image, then they must be the same element.
  • Again, every element from the domain have exactly one image in the codomain, and no two elements of the domain have the same image.
  • 

• Surjective Function, Surjection:

  • A function f : A → B is surjective if ∀b ∈ B, ∃a ∈ A such that f(a) = b.
  • Also called onto function.
  • That is, every element of the codomain is the image of at least one element of the domain.
  • Again, No one element of the codomain is not used as an image.
  • Again, every element from the codomain is connected to at least one element from the domain.
  • 

• Bijective Function, Bijection:

  • A function f : A → B is bijective if it is both injective and surjective.
  • Also called one-to-one, onto functions.
  • That is, every element of the codomain is the image of exactly one element of the domain, and every element of the codomain is used, and used for only one element of the domain.
• Cardinality. Two sets must have the same cardinality if the relationship between them is bijective.
• Countable Set: A set is countable if it is finite or if it has the same cardinality as some set of positive integers.
• Countably Infinite Set: A set is countably infinite if it has the same cardinality as the set of natural numbers N.
• Pigeonhole Principle:
  • If you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.
  • If n is a positive integer and n + 1 objects are placed into n boxes, then at least one box contains two or more objects.

Property	Meaning	English	English * (approximation, not very accurate)
Injective	\(∀a, b ∈ A, a ≠ b ⇒ f(a) ≠ f(b)\)	All elements in the domain have unique images in the codomain	All elements in the domain are used (unique images)
Surjective	\(∀b ∈ B, ∃a ∈ A, f(a) = b\)	All elements in the codomain have at least one image in the domain	All elements in the codomain are used

• Example, let \(f,g: R -> R\) where \(f(x) = \sqrt[2]{x}\) and \(g(x) = \sqrt[3]{x}\)

	\(f(x) = \sqrt[2]{x}\)	\(g(x) = \sqrt[3]{x}\)
Injective	No, Not all elements have unique images; e.g. negative numbers have no image.	Yes, all elements have unique cube root.
Surjective	Yes, negative numbers in codomain are used. e.g. \(f(16)= \pm2\)	Yes, all elements in codomain are used.
Bijective	No, not injective.	Yes, Injective and surjective

7.3 Function Composition

• Function Equality:
  • Two functions f and g are equal if they have the same domain and the same codomain and f(x) = g(x) for all x in the domain.
• Function Composition:
  • The composition of two functions f and g is the function f ◦ g defined by (f ◦ g)(x) = f(g(x)).
  • Example: f(x) = x^2 and g(x) = x + 1 then (f ◦ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2.
  • Also: f ◦ g ≠ g ◦ f as in the previous example (g ◦ f)(x) = g(f(x)) = g(x^2) = x^2 + 1.
• Function Composition Features is associative: (f ◦ g) ◦ h = f ◦ (g ◦ h).
• Powers of functions:
  • f^n = f ◦ f^(n-1) for n >= 1.
  • f^1 = f.
  • f^n+1 = f ◦ f^n.
• The composition of two injections is an injection: f: A → B and g: B → C are injections, then g ◦ f: A → C is an injection.
• The composition of two surjections is a surjection: f: A → B and g: B → C are surjections, then g ◦ f: A → C is a surjection.
• Identity Function:
  • It is a function that maps each element of a set to itself.
  • It is an onto function.
  • Composing any function with the identity function does not change the function: f ◦ I = I ◦ f = f.

7.3.3 Inverse Functions

• The inverse of a function f: A → B is a function g: B → A such that g ◦ f = I and f ◦ g = I.
• A function can only have one inverse or no inverse.
• The inverse function undoes the action of the original function.
• Example:
  • f: A → A where A={1,2,3} and f={(1,2), (2,3), (3,1)} .
  • The f^-1 = {(2,1), (3,2), (1,3)}. where we swapped the first and second components of each pair in f.
• So an inverse function is a function that takes the image of an element, and returns back the original element.
• Example:
  • f(x) = x^2 and g(x) = √x then f ◦ g = I and g ◦ f = I.
  • f ◦ g = f(g(x)) = f(√x) = (√x)^2 = x.
  • g ◦ f = g(f(x)) = g(x^2) = √(x^2) = x.
• Bijection functions are invertible.

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² Sequences

• A sequence is simply an ordered list of numbers. For example, here is a sequence: 0, 1, 2, 3, 4, 5, . . . ..
• The order of the element in the sequence is important.

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References

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1. Doerr, A., & Levasseur, K. (2022). Applied discrete structures (3^rd ed.). licensed under CC BY-NC-SA. https://discretemath.org/ads/chapter_7.html ↩

2. Levin, O. (2021). Discrete mathematics: An open introduction (3^rd ed.). licensed under CC 4.0. Chapter 2: Sequences. ↩