Discrete Structures

Lecture Notes, 5 Nov 2010

Last Update: 5 November 2010

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§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)
        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)
        {(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
        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!

  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!)

Next lecture…

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