Discrete Structures

Lecture Notes, 1 Nov 2010

 Last Update: 1 November 2010, 8:24 P.M. Note: or material is highlighted

§2.3: Functions

1. Reminder: Last time, I introduced some notation that cannot easily be shown in HTML; I'll do my best.

1. Notation:

i = n
∪Ai =def A1 ∪ A2 ∪ … ∪ An
i = 1

= (alternative notation)

n
∪Ai
i = 1

= (alternative notation)

n
∪Ai
1

= (alternative notation) i ∈ {1,…, n}Ai = (alternative notation) iAi

• (which should remind you of ∃iA(i))

2. E.g.:

Suppose (∀i ∈ {1,…, n})[Ai = {i}]
∴ A1 = {1}, A2 = {2}, …

1. Then:

4
∪Ai = {1} ∪ {2} ∪ {3} ∪ {4} = {1,2,3,4}
i=1

and

n
∪Ai = {1,2,3,…, n}
i=1

and

∞
∪Ai = iWAi = {1} ∪ … ∪ {i} ∪ … = W
i=1

3. Similarly:

i=n
∩Ai = A1 ∩ … ∩ An
i=1

• E.g.:

i=4
∩Ai = {1} ∩ {2} ∩ {3} ∩ {4} = ∅
i=1

2. Be sure to read "Using Sets to Define the Natural Numbers",

3. Functions:

• The book defines "function" in terms of an "assignment";
but "assignment" is never defined!

• Nor does the book define "function" in terms of "relations";
but I will :-)

2. Relations:

1. Informally, a binary relation between 2 objects is like a property that belongs to the ordered pair of the objects

E.g.:

1. x < y (e.g., 1 < 2)

• "<" is a 2-place relation between x & y
(or: it's a property of (x, y) )

2. x is a sibling of y (Kim is Pat's sibling)

• "being a sibling of" is a 2-place relation between x & y

3. x is a course that is taken by student y (191 is a course that you take)

4. x is between y and z (2 is between 1 and 3)

2. (Formal, Set-Theoretic) Definition:

Let A,B be sets.
Then R is a binary relation on A × B
or: R is a binary relation from A to B
or: R is a binary relation between A and B

=def R ⊆ A × B

• i.e., R ⊆ {(a, b) | (a ∈ A) ∧ (b ∈ B)}

3. Functions:

1. Basic idea: A function is a relation s.t. the same input (I/P) always has the same output (O/P)

• So, the graphs of curves like a straight line with slope = 1, or a parabola open at the top, or a sine wave are functions.
• But a circle is not a function.
• And √ is not a function (each input has two outputs).

2. Def:

Let A,B be sets.
Then f is a function from A to B

=def

1. f is a binary relation from A to B,
and
2. (∀a∈A)(∀b∈B)(∀b′∈B)[( (a,b) ∈ f  ∧  (a,b′) ∈ f ) → b=b′]

3. i.e., no two distinct members of f have the same first element (but different 2nd elements)

4. I.e., if "2" members of f have the same first element,
then they have the same second element.

• i.e., it only seems as if they are 2 members.
• i.e., "they" are one, not two

5. i.e., same I/P → same O/P

4. Notation & Terminology:

1. "f : A → B" for: f is a function from A to B

• Note: The "→" is NOT a material-conditional arrow!!!!!

2. "f(a) = b"
or                      for: (a, b) ∈ f
"f : a |→ b"

3. In the above, "a" and "b" are associated with different jargon in different academic disciplines:

disciplineab
social sciencesindependent
variable
dependent
variable
b is a function of a
b depends on a
mathematicspre-image of b image of a
computer scienceinputoutput

4. f "maps" A "to" B

f is a "transformation of" A "into" B

5. Def:

Let A,B be sets.
Let a,c ∈ A; b,d ∈ B.
Let f, g : A → B
Then f = g  =def  {(a, b) | (a, b) ∈ f} = {(c, d) | (c, d) ∈ g}

1. i.e., f = g  ↔  ∀ab[(a, b) ∈ f  ↔  (a, b) ∈ g]

2. i.e., f = g  ↔  ∀ab[f(a) = b  ↔  g(a) = b]

3. i.e, "2" functions are the same iff they are the same set

6. The So-Called "Function Machine"

1. Consider a machine f that:

• takes input a,
• you turn a crank,
• it grinds away at the input,
• and finally it outputs b (i.e., f(a)):

2. Despite what you may have been told elsewhere (e.g., in high school), this is NOT what a function is!

3. A function is merely the set of input-output (I/O) pairs.

4. What the machine really is is a computer!

• (And the "gears" are an algorithm that computes the function.)

• But: not all functions can be computed by algorithms!
• I.e., there are functions for which there are no such "function machines"