Recitation 6: Functions and Set Operations

Part 1: Set Operations

Let A = {a, b, c} and B = {x, y, z}

• What is A ∪ B?
• What is A ∩ B?
• What is the Cartesian product of A and B?
• Identity Laws
  • Set Difference Law: A − B = A ∩ B^c

Part 2: Functions

• Describing functions as sets of tuples
  • domain = D
  • target = T
  • range = R
  • f = {(element of D, element of T),(element of D, element of T),...} ⊆ D × R ⊆ D × T
• Function Validity
  • are these functions valid? (diagrams will be drawn on the board)
  • what is the vertical line test and how can we use it on these problems?
• Injective functions: how to tell if a function is injective
  • Let f : ℝ → ℝ be defined by f(x)=5x + 1. Is f(x) injective?
  • Let g : ℝ → ℝ be defined by g(x)=|x|. Is g(x) injective?
  • what is the horizontal line test and how can we use it on these problems?
• Surjective functions: how to tell if a function is surjective. All the following functions are ℝ → ℝ. Which are surjective?
  • h(x)=5x + 1
  • j(x)=sin(x)
  • k(x)=2^x