cmps 2433 chapter 2 – part 2 functions & relations
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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 - PowerPoint PPT PresentationTRANSCRIPT
HALVERSON – MIDWESTERN STATE UNIVERSITY
CMPS 2433 Chapter 2 – Part 2
Functions & Relations
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.}
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.
Relations (cont’d)
Example: R is a relation on AA = {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
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?
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}
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}
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}
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}
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}
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
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}
Homework on Relations - Section 2.2
Page 52 – 54Problems 1 – 14, 19-20, 25
Section 2.4 - Functions
A Function f 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)}
Mathematical Functions
Consider mathematical FUNCTIONSAssume 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!
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}
Function Domain
If f is a function from X to Y, denote f: X YSet X is called the domain of the
functionSet Y is called co-domain Subset of Y actually paired with
elements of X under f is called the range
For f(x) = y, y is the image of x under f
Domain, Co-domain, Range Examples
S = {…, -3,-2,-1,0, 1, 2, 3, 4, 5,…}Define f as a function on SF(x) = x2
Domain = SCo-domain = SRange = ???
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?
Exponential & Logarithmic Functions
Logarithmic functions IMPT in Computing
Generally, base 2NOTE:
2n is exponential function, base 220 = 1 and 2-n = 1/2n
See page 73 for graph – Figure 2.18Logarithmic Function base 2 is inverse
of Exponential Function
Logarithmic Function - Base 2
Notation: log2 x Read “log base 2 of x”
Defn: y = log2 x if and only if x = 2y
Examples: log2 8 = 3 because 23 = 8 log2 1024 = 10 because 210 = 1024 log2 256 = 8 because 28 = 256 log2 100 ~~ 6.65 because 26.65 ~~ 100
More on Logarithmic Function - Base 2
Growth rate is small, less than linearSee graph page 74 – Figure 2.19Calculator Note:
Most calculators with LOG button is base 10
log 2 x = LOG x / LOG 2
Algorithms with O(log2 n) complexity??
Homework – Section 2.4
Note – we omitted section on Composite & Inverse Functions
Page 74-75Problems 1 - 36