Functions - Discrete Structures - Lecture Slides, Slides of Discrete Structures and Graph Theory

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… and the following mathematical

appetizer is about…

• Functions

Functions

• A function f from a set A to a set B is an assignment of exactly one element of B to each element of A.
• We write
• f(a) = b
• if b is the unique element of B assigned by the function f to the element a of A.
• If f is a function from A to B, we write
• f: A→B
• (note: Here, “→“ has nothing to do with if… then)

Functions

• Let us take a look at the function f:P→C with
• P = {Linda, Max, Kathy, Peter}
• C = {Boston, New York, Hong Kong, Moscow}
• f(Linda) = Moscow
• f(Max) = Boston
• f(Kathy) = Hong Kong
• f(Peter) = New York
• Here, the range of f is C.

Functions

• Let us re-specify f as follows:
• f(Linda) = Moscow
• f(Max) = Boston
• f(Kathy) = Hong Kong
• f(Peter) = Boston
• Is f still a function? yes

What is its range? {Moscow, Boston, Hong Kong}

Functions

• If the domain of our function f is large, it is convenient to specify f with a formula, e.g.:
• f: R → R
• f(x) = 2x
• This leads to:
• f(1) = 2
• f(3) = 6
• f(-3) = -
• …

Functions

• Let f 1 and f 2 be functions from A to R.
• Then the sum and the product of f 1 and f 2 are also functions from A to R defined by:
• (f 1 + f 2 )(x) = f 1 (x) + f 2 (x)
• (f 1 f 2 )(x) = f 1 (x) f 2 (x)
• Example:
• f 1 (x) = 3x, f 2 (x) = x + 5
• (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) = 3x + x + 5 = 4x + 5
• (f 1 f 2 )(x) = f 1 (x) f 2 (x) = 3x (x + 5) = 3x 2 + 15x

Functions

• Let us look at the following well-known function:
• f(Linda) = Moscow
• f(Max) = Boston
• f(Kathy) = Hong Kong
• f(Peter) = Boston
• What is the image of S = {Linda, Max}?
• f(S) = {Moscow, Boston}
• What is the image of S = {Max, Peter}?
• f(S) = {Boston}

Properties of Functions

• A function f:A→B is said to be one-to-one (or injective), if and only if
• ∀x, y∈A (f(x) = f(y) → x = y)
• In other words: f is one-to-one if and only if it does not map two distinct elements of A onto the same element of B.

Properties of Functions

• How can we prove that a function f is one-to-one?
• Whenever you want to prove something, first take a look at the relevant definition(s):
• ∀x, y∈A (f(x) = f(y) → x = y)
• Example:
• f: R → R
• f(x) = x 2
• Disproof by counterexample:
• f(3) = f(-3), but 3 ≠ -3, so f is not one-to-one.

Properties of Functions

• … and yet another example:
• f: R → R
• f(x) = 3x
• One-to-one: ∀x, y∈A (f(x) = f(y) → x = y)
• To show: f(x) ≠ f(y) whenever x ≠ y
• x ≠ y

⇔ 3x ≠ 3y

⇔ f(x) ≠ f(y),

so if x ≠ y, then f(x) ≠ f(y), that is, f is one-to-one.

Properties of Functions

• A function f:A→B is called onto, or surjective, if and only if for every element b∈B there is an element a∈A with f(a) = b.
• In other words, f is onto if and only if its range is its entire codomain.
• A function f: A→B is a one-to-one correspondence, or a bijection, if and only if it is both one-to-one and onto.
• Obviously, if f is a bijection and A and B are finite sets, then |A| = |B|.

Properties of Functions

• Examples:
• In the following examples, we use the arrow representation to illustrate functions f:A→B.
• In each example, the complete sets A and B are shown.

Properties of Functions

• Is f injective?
• No.
• Is f surjective?
• Yes.
• Is f bijective?
• No.

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

Paul

Properties of Functions

• Is f injective?
• Yes.
• Is f surjective?
• No.
• Is f bijective?
• No.

Linda

Max

Kathy

Peter

Boston

New York

Hong Kong

Moscow

Lübeck