Introduction to Functions

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• Questions?
• Function Terminology
  • Section 6.1
  • Suppose f(n) = 0 if n is an even integer, and f(n) = 1 if n is an odd integer
    • What does the domain of f seem to be?
      • Domain is possible n’s
      • So domain = ℤ
    • What are some sets that could be codomains of f?
      • Codomain is output
      • So codomain = {0,1}
      • Or ℤ
    • How would you specify f to get rid of “seem to be” and “could be” re domain and codomain?
      • f : ℤ → ℤ or f : ℤ → {0,1} etc.
    • What is the range of f?
      • Range is just the outputs
      • So range = {0,1}
    • What is the image of 2?
      • Image of x is what x maps to
      • So image of 2 = 0
    • What is the set of preimages of 1?
      • Preimage of y is what maps to y
      • So preimage of 1 = { …, -3, -1, 1, 3, … }
  • Progress check 6.4 (arrow diagrams)
  • Is this a description of a function: { (0,0), (1,1), (2,4), (3,9), (4,16) }?
    • f(x) = x²?
    • Or a more accurately f : {0,1,2,3,4} → {0,1,4,9,16} where f(x) = x²
    • Function f is subset of A × B
      • “f maps every element of its domain”: for all a ∈ A, (a,b) ∈ f for some b ∈ B
      • “every element of domain maps to exactly 1 element of codomain”
        • For all a ∈ A, (a,b) ∈ f for some unique b ∈ B
        • Or more usually for all x and y ∈ B and for all a ∈ A, f(a) = x and f(a) = y iff x = y
  • Problem set
• Next
  • Function equality
    • Read section 6.2

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