Logical structure 2 show that the hypotheses it is

Info icon This preview shows pages 48–60. Sign up to view the full content.

View Full Document Right Arrow Icon
LOGICAL STRUCTURE 2. Show that the hypotheses "It is not sunny this afternoon and it is colder than yesterday," "We will go swimming only if it is sunny," "If we do not go swimming, then we will take a canoe trip, " and " If we take a canoe trip, then we will be home by sunset" lead to the conclusion "We will be home by sunset. "
Image of page 48

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
FUNCTIONS
Image of page 49
FUNCTIONS Definition Let A and B be sets. A function f from A to B is an assignment of exactly one B element to each element of A . We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A. If is a function from A to B , we write f:A→B .
Image of page 50

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
FUNCTIONS Definition If f is a function from A to B , we say that A is the domain of f and B is the codomain of f . If f(a)=b , we say that b is the image of a and a is a pre-image of b . The range of f is the set of all images of elements of A . Also, if f is a function from A to B , we say that f maps A to B .
Image of page 51
FUNCTIONS A function f from A to B is said to be one- to-one , or injective if for each b ϵ B, there is at most one a A with f(a)=b ϵ . A function is said to be an injection if it is one-to-one. A function f from A to B is called onto , or surjective if and only if for every element b B ϵ there is an a A ϵ element with f(a)=b . A function is said to be surjection if it is onto.
Image of page 52

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
FUNCTIONS The function f is a one-to-one correspondence , or bijection , if it is both one-to-one and onto. Example:Let S={1,2,3,4,5} and T={a,b,c,d}. For each question below if your answer is yes , give an example; if your answer is no , explain briefly. 1. Are there any one-to-one functions from S to T? 2. Are there any functions mapping S onto T?
Image of page 53
FUNCTIONS Definition(Inverse) Let f be a one-to-one correspondence from the set A to the set B . The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b . The inverse function of f is denoted by f¯¹ . Hence, f¯¹ (b)=a when f(a)=b .
Image of page 54

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
FUNCTIONS Definition(Composition) Let g be a function from the set A to the set B and let f be a function from the set B to the set C . The composition of the functions f and g , denoted by f◦g is defined by (f◦g)(a)=f(g(a)) . (not defined unless the range of g is a subset of the domain of f)
Image of page 55
FUNCTIONS Examples: Let A = {1,2,3,4,5} B ={6,7,8,9} C= {10,11,12,13} D ={□,∆,◊,∂} Let R AxB, S BxC and T CxD be defined by R = {(1,7),(4,6),(5,6),(2,8)} S = {(6,10),(6,11),(7,10),(8,13)} T = {(11, ∆),(10, ∆),(13, ∂),(12, □),(13, ◊)} Compute the relations
Image of page 56

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
FUNCTIONS Definition The floor function that assigns to the real number x is the largest integer that is less than or equal to x . The value of the floor function at is denoted by Lx . The ˩ ceiling function assigns to the real number x the smallest integer that is greater than or equal to x . The value of the ceiling function at is denoted by Гx . ˥
Image of page 57
COMBINATORIAL STRUCTURES
Image of page 58

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
COUNTING Examples 1.
Image of page 59
Image of page 60
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern