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

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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 .

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