Lesson 6: Functions

1
Lesson 6 Functions

• Objectives
• Use bit strings to represent sets
• Define the terms in Section 1.8 up to page 101
• Determine if a function is one-to-one

• Outline
• Bit String Representation
• Functions
• Terms
• Sets
• Images
• One-to-One functions
• Reading Section 1.8

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Bit String Representation of Sets

• Define ordering of all possible members of the
  set
• Create corresponding bit string
• 0 represents no membership
• 1 represents membership
• Set operations then easily performed as bit
  manipulations
• Does not work where the Universe is infinite

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Bit String Representation of Sets

• Universe apple, orange, peach, guava,
  nectarine, mango, grape, banana
• S grape, apple, peach
• T peach
• U banana, guava, nectarine, apple, orange
• S?T
• S - U

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Functions

• A function maps each member of a set uniquely to
  a member of a second set
• f A?B
• f(a) b
• Function Representations
• Explicit
• Formula

5
Explicit Function Definition

• S edible, non-edible
• T steel, tofu, polyethylene, ethanol, cookies
  

6
Functions

• Consider a function f from A to B
• A domain of f
• B codomain of f
• f(a) b
• b is the image of a
• a is the pre-image of b
• Set of all possible images range of f
• f maps A to B

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Function

• f Z ? Z, f is the square of the integer
• Domain
• Codomain
• Image of (-4)
• Range

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Function

• f ? ? ?, f is the square of the number
• Domain
• Codomain
• Image of (-4)
• Range

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Functions and Sets

• If f1,f2 ? ? ?
• f1f2 ? ? ?
• f1 f2 ? ? ?
• Let f be a function from A to B, and let S ? A
• Image of S is a subset of B
• Image of S contains images of the elements of S
• Denoted f(S)
• F(S) f(s) s ? S

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Set Image

• A Bill, Susie, Ed, Ann
• B A, B, C, D, F
• Bill got an A, Susie got a C, Ed got a C, Ann got
  a B
• S boys in A
• Range of f
• f(S)

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One-to-One Functions

• A function f is one-to-one iff each element maps
  uniquely to the domain
• Also called an injection
• A A, C, D, E, T
• B 1, 2, 3, 4, 5, 6, 20
• f(A) 1, f(C) 3, f(D) 4, f(E) 5, f(T) 20
• f(x) x4, Z?Z

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Halide

• A function from reals to reals for which f(x) gt
  f(y) whenever x gt y is called strictly increasing
• Strictly decreasing
• Increasing
• Decreasing
• An onto function f A ? B maps at least one
  element of A to every element of B
• surjective