1 / 23

CMPS 2433 Chapter 2 – Part 2 Functions & Relations

CMPS 2433 Chapter 2 – Part 2 Functions & Relations. Halverson – Midwestern State University. 2.2 Relations. A RELATION from set A to set B is any subset of the Cartesian Product A X B If R is a relation from A to B & (a, b) is an element of R, a is related to b by R Example

laith-rhodes
Download Presentation

CMPS 2433 Chapter 2 – Part 2 Functions & Relations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CMPS 2433 Chapter 2 – Part 2Functions & Relations Halverson – Midwestern State University

  2. 2.2 Relations • A RELATION from set A to set B is any subset of the Cartesian Product A X B • If R is a relation from A to B & (a, b) is an element of R, a is related to b by R • Example • A = {students enrolled at MSU in fall 2014} • B = {courses offered at MSU in fall 2014} • R = {(a,b)| student a is enrolled in course b} • R = {(Smith,Math1233), (Jones, CMPS1044), etc.}

  3. Relations (cont’d) • A relation is ANY subset, so no repeated pairs but can have repeated elements in the pairs. R = {(Smith,Math1233), (Jones, Cmps1044), (Smith, Engl1013), (Jones, Math1233), (Hunt, Math1233), (Williams, Cmps1044), etc.} What is the Universal Set for R? Define a different Relation from A to B.

  4. Relations (cont’d) Example: R is a relation on A A = {students enrolled at MSU in fall 2014} R = {(a, b)| a & b are in a course together} R = {(Smith, Jones), (Jones, Hunt), (Hunt, Wills), (Wills, Johnson), etc.} What about (Jones, Smith)? Relation from a set S to itself is call a Relation on S

  5. Reflexive Property of Relations • A Relation R on a set S is said to be Reflexive • if for each x  S, x R x is true • if for each x  S, (x, x) is in R • that is, every element is related to itself • Is our previous example R a Reflexive Relation?

  6. Reflexive Property of Relations - Examples A = {students enrolled at MSU in fall 2014} Which of the following are Reflexive? • R = {(a,b): a & b are siblings} • R = {(a,b): a & b are not in a course together} • R = {(a,b): a & b are same classification} • R = {(a,b): a & b are married} • R = {(a,b): a & b are the same age} • R = {(a,b): a has a higher GPA than b}

  7. Symmetric Property of Relations • A Relation R on a set S is said to be Symmetric • If x R y is true, then y R x is true • If (x, y)  R, then (y, x) is true • That is, the elements of the relation R can be reversed Is R Symmetric? R = {(a, b)| a & b are in a course together}

  8. Symmetric Property of Relations - Examples A = {students enrolled at MSU in fall 2014} Which of the following are Symmetric? • R = {(a,b): a & b are siblings} • R = {(a,b): a & b are not in a course together} • R = {(a,b): a & b are same classification} • R = {(a,b): a & b are married} • R = {(a,b): a & b are the same age} • R = {(a,b): a has a higher GPA than b}

  9. Transitive Property of Relations • A Relation R on a set S is said to be Transitive • If x R y and y R z are true, then x R z is true • If (x, y) R & (y, z) R, then (x, z) R Is R Transitive? R = {(a, b)| a & b are in a course together}

  10. Transitive Property of Relations - Examples A = {students enrolled at MSU in fall 2014} Which of the following are Transitive? • R = {(a,b): a & b are siblings} • R = {(a,b): a & b are not in a course together} • R = {(a,b): a & b are same classification} • R = {(a,b): a & b are married} • R = {(a,b): a & b are the same age} • R = {(a,b): a has a higher GPA than b}

  11. Equivalence Relation • Any Relation that is Reflexive, Symmetric & Transitive is an Equivalence Relation • If R is an Equivalence Relation on S & x  S, the set of all elements related to x is called an Equivalence Class • Denoted [x] • Any 2 Equivalence Classes of a Relation are either Equal or Disjoint • The Equivalence Classes of R Partition S

  12. Equivalence Relations - Examples A = {students enrolled at MSU in fall 2014} Which are Equivalence Relations? If so, what are the partitions? • R = {(a,b): a & b are siblings} • R = {(a,b): a & b are not in a course together} • R = {(a,b): a & b are same classification} • R = {(a,b): a & b are married} • R = {(a,b): a & b are the same age} • R = {(a,b): a has a higher GPA than b}

  13. Homework on Relations - Section 2.2 Page 52 – 54 Problems 1 – 14, 19-20, 25

  14. Section 2.4 - Functions A Functionf from set X to set Y is a relation from X to Y in which for each element x in X there is exactly one element y in Y for which x f y Among the ordered pairs (x, y) in f, x appears only ONCE Example: is F a function? F = {(2,3), (3,2), (4,2)} F = {(2,3), (3,2), (2,4), (4,6)}

  15. Mathematical Functions Consider mathematical FUNCTIONS Assume S = {0, 1, 2, 3, 4, 5,…} f(x) = x2 = {(0,0), (1,1),(2,4),(3,9),(4,16),…} f(x) = x+2 = {(0,2),(1,3),(2,4),(3,5),…} For every x, there is only ONE value to which it is related, thus these are Functions!

  16. Equivalence Relations - Examples A = {students enrolled at MSU in fall 2014} Which are Functions? • R = {(a,b): a & b are siblings} • R = {(a,b): a & b are not in a course together} • R = {(a,b): a & b are same classification} • R = {(a,b): a & b are married} • R = {(a,b): a & b are the same age} • R = {(a,b): a has a higher GPA than b}

  17. Function Domain If f is a function from X to Y, denote f: X  Y Set X is called the domain of the function Set Y is called co-domain Subset of Y actually paired with elements of X under f is called therange For f(x) = y,y is the image of x under f

  18. Domain, Co-domain, Range Examples S = {…, -3,-2,-1,0, 1, 2, 3, 4, 5,…} Define f as a function on S F(x) = x2 Domain = S Co-domain = S Range = ???

  19. Functions – additional terms • One-to-One function • For every x, there is a unique y & • For every y, there is a unique x • {(x, y)| no repeats of x or y} S = {…, -3,-2,-1,0, 1, 2, 3, 4, 5,…} f(x) = x2 Is f one-to-one?

  20. Exponential & Logarithmic Functions • Logarithmic functions  IMPT in Computing • Generally, base 2 • NOTE: • 2n is exponential function, base 2 • 20 = 1 and 2-n = 1/2n • See page 73 for graph – Figure 2.18 • Logarithmic Function base 2 is inverse of Exponential Function

  21. Logarithmic Function - Base 2 • Notation: log2 x • Read “log base 2 of x” • Defn: y = log2x if and only if x = 2y • Examples: • log2 8 = 3 because 23 = 8 • log21024 = 10 because 210= 1024 • log2256 = 8 because 28= 256 • log2100 ~~ 6.65 because 26.65 ~~ 100

  22. More on Logarithmic Function - Base 2 • Growth rate is small, less than linear • See graph page 74 – Figure 2.19 • Calculator Note: • Most calculators with LOG button is base 10 • log 2 x = LOG x / LOG 2 • Algorithms with O(log2 n) complexity??

  23. Homework – Section 2.4 Note – we omitted section on Composite & Inverse Functions Page 74-75 Problems 1 - 36

More Related