Discrete Structures

# Lecture Notes, 5 Nov 2010

 Last Update: 5 November 2010 Note: or material is highlighted

## §2.3: Functions (cont'd)

1. 1–1 Correspondence:

1. Reminders:

1. function f : A → B
• same I/P → same O/P

2. 1–1 function f : A → B
• (∀a,a′∈A)[f(a)=f(a′) → a=a′]
• or: aa′ → f(a)≠f(a′)
• i.e.) same O/P → same I/P
• i.e.) different I/P → different O/P

3. onto function f : A → B
• (∀b∈B)(∃a∈A)[f(a)=b]

• i.e.) every element of co-domain is in range
• or: every "target" is hit by some "archer"

4. Now we put these together:

2. Def:

Let A,B be sets.
Let f  : A → B be a total function
Then f is a 1–1 correspondence between A and B
(or:      is a bijection)
=def
f  is 1–1 and onto.

3. E.g., the identity function
ιA : A → A s.t. (∀a ∈ A)[ιA(a) = a]
is a 1–1 correspondence

2. Inverses of Functions:

1. Def:

Let A,B be sets.
Let f  : A → B.
then the inverse of f, denoted f–1 (a relation from B → A)
=def
{(b, a) | (a, b) ∈ f }

2. Now, f  is a function (by hypothesis).
But is f–1 a function, or merely a non-functional relation?

• I.e., can we write: f–1(b) = a  ↔  f(a) = b?

3. Thm (You Can Go Home Again):

Let A,B be sets.
Then relation f–1 : B → A is a total function
iff
f  : A → B is a 1–1 correspondence.

proof sketch:

Case 1: Show f–1 is a total function → f is 1–1:

Use proof by contrapositive:
f not 1–1 → ∃b ∈ B that is image of >1 a ∈ A
∴ f–1 not a function;
i.e., you can't go home again, because you don't know which is your home.

Case 2: Show f–1 is a total function → f is onto:

Another proof by contrapositive:
f not onto → ∃b ∈ B that is not image of any a ∈ A;
i.e., you can't go home again, because there's no home to go to.

Case 3: Show f is a 1–1 correspondence → f–1 is a total function:

Left up to you!
QED

3. Function Composition:

1. ∃ operations on functions:

• i.e., ∃ an "algebra" of functions

2. Def:

Let A,B,C be sets.
Let g : A → B.
Let f : B → C.
Then the composition of f with g
—denoted "(f o g) : A → C" & read "f of g"—
=def {(a, c) | (∃b ∈ B)[(g(a) = b) ∧ (f(b) = c)]

1. i.e., (f o g)(a) = f(g(a))

2. E.g.:

Let f(x) = x+1.
Let g(y) = 3y.

What are f o g and g o f? (answer next time!)