2.4 - 2-39 2.4 Sequences and Summations 38 d x y = x y for...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
2-39 2.4 Sequences and Summations 38 d) xy = x y for all real numbers x and y . e) x 2 = x + 1 2 for all real numbers x . 70. Prove or disprove each of these statements about the floor and ceiling functions. a) x = x for all real numbers x . b) x + y = x + y for all real numbers x and y . c) x / 2 / 2 = x / 4 for all real numbers x . d) x = x for all positive real numbers x . e) x + y + x + y 2 x + 2 y for all real numbers x and y . 71. Prove that if x is a positive real number, then a) x = x . b) x = x . 72. Let x be a real number. Show that 3 x = x + x + 1 3 + x + 2 3 . A program designed to evaluate a function may not produce the correct value of the function for all elements in the domain of this function. For example, a program may not produce a correct value because evaluating the function may lead to an infinite loop or an overflow. Similarly, in abstract mathemat- ics, we often want to discuss functions that are defined only for a subset of the real numbers, such as 1 / x , x , and arcsin ( x ). Also, we may want to use such notions as the “youngest child” function, which is undefined for a couple having no children, or the “time of sunrise,” which is undefined for some days above the Arctic Circle. To study such situations, we use the concept of a partial function. A partial function f from a set A to a set B is an assignment to each element a in a subset of A , called the do- main of definition of f , of a unique element b in B . The sets A and B are called the domain and codomain of f , respec- tively. We say that f is undefined for elements in A that are not in the domain of definition of f . We write f : A B to denote that f is a partial function from A to B . (This is the same notation as is used for functions. The context in which the notation is used determines whether f is a partial function or a total function.) When the domain of definition of f equals A , we say that f is a total function. 73. For each of these partial functions, determine its domain, codomain, domain of definition, and the set of values for which it is undefined. Also, determine whether it is a total function. 74. a) Show that a partial function from A to B can be viewed as a function f from A to B ∪ { u } , where u is not an element of B and f ( a ) = f ( a ) if a belongs to the domain of definition of f u if f is undefined at a . b) Using the construction in (a), find the function f cor- responding to each partial function in Exercise 73. 75. a) Show that if a set S has cardinality m , where m is a positive integer, then there is a one-to-one correspon- dence between S and the set { 1 , 2 ,..., m } . b) Show that if S and T are two sets each with m ele- ments, where m is a positive integer, then there is a one-to-one correspondence between S and T .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
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