Lecture Notes, 5 Nov 2010

Note: A username and password may be required to access certain documents. Please contact Bill Rapaport.

Index to all lecture notes
…Previous lecture

§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: a≠a′ → 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
      ι_A : A → A s.t. (∀a ∈ A)[ι_A(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. 
         

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

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

Next lecture…