W13:Homework:H07. CS40 Foundations of Computer Science W13. From 40wiki

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1 W13:Homework:H07 From 40wiki CS40 Foundations of Computer Science W13 W13:Exams W13:Homework in Class and Web Work W13:Calendar W13:Syllabus and Lecture Notes UCSB-CS40-W13 on Facebook ( Next in the Array of Talks Series: Talks[5]: From Academia to Business: All about startups! ( Wednesday Jan 23, 3pm-5pm click above to learn more and/or to RSVP on Facebook! CS40 on Gauchospace ( id=5907) H01 H02 H03 H04 H05 H06 H07 H08 H09 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 W01 W02 W03 W04 W05 W06 W07 W08 W09 W10 IC01 IC02 IC03 IC04 IC05 IC06 IC07 IC08 IC09 IC10 IC11 IC12 [Print-friendly PDF ( ]

2 H07-W13-CS40 page 1 First name (color-in initial) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z section (1 or 4) first name initial last name initial Last name (color-in initial) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z H07: Due Wednesday in Lecture. Total Points: 50 MAY ONLY BE TURNED IN DURING Lecture ON Wednesday , or offered in person, for in person grading, during instructor or TAs office hours. See the course syllabus at for more details. (1) (10 pts) Fill in the information below. Also, fill in the A-Z header by coloring in the first letter of your first and last name (as it would appears in Gauchospace), writing either 1 or 4 to indicate your discussion section meeting time, i.e., Tuesday at 1pm or Tuesday at 4pm. writing your first and last initial in large capital letters (e.g. P C). All of this helps us to manage the avalanche of paper that results from the daily homework. name: umail Your reading assignment for H07 includes Sections 2.1, 2.2, 2.3 from your textbook The handout given out with this assignment (which can be found on the wiki under the link for Homework H07---it is part of the printable PDF) (2) (20 pts) For each of the graphs below that shows a mapping from elements of set A to set B indicate (by circling yes or no) whether or not the mapping is a function f: A B. If you say no, explain your answer. Graph: Is it a function? yes no yes no yes no yes no Explanation:

3 F12-H11-page 2 (3) (12 pts) A function f: A B is injective (or 1-to-1) if for each element x A, if f(x)=b, then y A, y x, f(y) = b. That is, it is not allowed for two different elements in A to both map to the same element in B. A function f: A B is surjective (or onto) if for each element b B, x A such that f(a)=b. That is, there is nothing in B that doesn't have something in A mapped to it. (No lonely elements in B! Remember it this way: the mapping is "onto" the entire B set. For each of the mappings below, indicate whether it is a function, whether it is 1-to-1 and whether it is onto or not. Graph: Is it a function? yes no yes no Is it 1-to-1? yes no yes no Is it onto? yes no yes no Graph: Is it a function? yes no yes no Is it 1-to-1? yes no yes no Is it onto? yes no yes no (4) (4 pts) Can f(x)=x 2 be considered a function f: R Z? Explain your answer. (5) (4 pts) What is a partial function, and what is the "hack" that can be used to turn a partial function into a function?

4 CS40-W13-H07 Handout Page 1 Handout to go with H07 This handout reviews basic material about functions. This is some of the most important material from Sections 2.1, 2.2 and 2.3 of the textbook. You should also read those sections of the book. Concentrate, in your first reading, on the material that is ALSO covered on this handout. You are responsible for all the material in Sections 2.1, 2.2 and 2.3 (unless you are told otherwise later) but the MOST important material from those sections pertains to the topics on this handout. Reading this handout is an INTRODUCTION, SUPPLEMENT and a GUIDE to reading those textbook sections, not a SUBSTITUTE. About Functions You are, no doubt, already familiar with the idea of a function from your study of algebra, trigonometry and calculus in your high school and college mathematics courses so far. For example, if we write: f(x)=x 2 then f(3) = 9 and f(0.5)=0.25 In the course, we are going to look at how the idea of a function is rooted in set theory. What you already know about functions will still hold true, but it is not the complete picture. It turns out that this way of looking at a function is the foundation of the idea of a function as it appears in programming languages such as C, C++, Python, JavaScript, etc. and the idea of a method in object-oriented languages such as Java. We may begin to see that in this course, and it may become more apparent in later courses such as CS138 and CS162. Functions from one set to another Suppose we have two sets, A and B, with A={s,t,u,v,w} and B={1,2,3} We can define a function that maps each element of A to exactly one element of B. Here is one such function, defined by enumerating each value of f(x) for every x A f(s) = 1 f(t) = 2 f(u) = 2 f(v) = 3 f(w) =3 We can also represent that function as a graph, with the elements of A on the left, and the elements of B on the right, as shown in the drawing at right labelled f. We could define another function g in the same way, that also maps each element of A to exactly one element of B. g(s) = 1 g(t) = 2 g(u) = 3 g(v) = 1 g(w) =2 Again, that can be shown as a graph see the graph at right. Here are several important things to know about functions: The notation f: A B is a way of writing that function f maps each element of the set A to exactly one element of the set B. I've been putting the words "each" and "exactly one element" in bold every time they appear to emphasize that these are two requirements of what it means to be a function. If some mapping f does not map "each" element of A, i.e. there is some element x A for which f(x) is not defined, then f is not a function* If some mapping maps some element x A to y B and also to z B, and y z, then f is not a function. (*It is not a function, though it fact it may be a different kind of thing called a partial function you'll learn about partial functions in the reading for Chapter 2, on p. 81).

5 CS40-W13-H07 HANDOUT page 2 Examples of things that are not functions Consider the mappings defined by the graphs shown at right, i.e. h(s) = 1 h(s) = 2 h(t) = 2 h(u) = 2 h(v) = 3 h(w) = 3 j(s) = 1 j(t) = 2 j(u) = 2 j(w) = 3 h is not a function because h(s) maps to both 1 and 2 in set B. This is not allowed. In a function that maps from A to B, for each element x A there must be one and only one value f(a) B. j is not a function because there is no value defined for j(v). This is not permitted for functions. To sum up: The definition of a function f that maps A to B, written f: A B, is that: x A, f(x) B x A, if f(x)=a then b B b a and f(x)=b Putting functions like f(x)=x 2 into this framework As a reminder, here are some traditional notations for certain sets of numbers, i.e. Z = {... -2, -1, 0, 1, 2,... }, the set of all integers N = {0,1,2,... }, the set of all natural numbers (our textbook author includes zero, though some authors don't) R = all real numbers (which as we will see, cannot be listed the same way that sets of integers can be) Note that N Z and Z R With this notation, the function f(x)=x 2 can be considered as any of the following: f : N N f : Z N f : Z Z f : R R However, the f: R Z is NOT a valid way of considering f(x)=x 2. Can you explain why? (I hope so, because I'm going to ask you as one of the in-class exercises. If you don't understand, ask one of your classmates for her/his thoughts on the matter, or as a last resort, ask your TA/Instructor to give you some hints. Functions like f(x)=x 2 as a "special case" of binary relations When a function has a single argument (i.e. it has arity 1, or it maps from a simple set to another set, rather than from a Cartesian product to another set), we can think of the function as a special case of a binary relation. Before we "go there", first consider the cartesian product R R. Every element of R R is an ordered pair (x,y), where x&isinr and y R. That notion should be familiar to you in fact, the set R R corresponds exactly to the idea of the Cartesian plane. This is sometimes called R 2, which is sometimes called "two space", short for "two dimensional space". Perhaps you can now see how the notions of "Cartesian product" and "Cartesian plane" relate to each other! Now, consider: S R R That is, S is a binary relation on the Reals, i.e. a subset of the Cartesian product of the set of Real numbers with itself. Suppose that: S R R, where S= {(x,y) R R y = x 2 } Now we can see that: S corresponds exactly to the function f(x)=x 2 If we plot every point that is an element of S in the Cartesian Plane, we get the plot at right.

6 CS40-W13-H07 HANDOUT page 3 More on functions as a special case of binary relations Please read Example 2.4 at the top of p. 80 before continuing. Did you read it? If so, thanks. If not, why not? I really mean it! Read it! So, my question to you is: Can every binary relation be thought of as a function? If not, then what would we need to show to prove that A given binary relation R is a function? A given binary relation R is NOT a function? Think about these questions before continuing--you may be asked about them later. If you are not confident of your answers, ask in lecture, discussion section, or office hours. Some terms/concepts you need to know from Sections 2.1, 2.2, 2.3: an INITIAL list There may be more items added to this list later. Review the definitions of the following important terms. function domain co-domain "type" of a function range (as different from co-domain) what the notation f: A B means what the expression A B means (without the f: in front) how a tuple can be thought of as a function how an infinite sequence can be thought of as a function how certain binary relations can be thought of as functions (i.e. how we can ask the question: is this binary relation a function, yes or no, and how we determine the answer.) defining a function by cases (e.g. absolute value) what a "partial function" is know an example of a partial function know what the "hack" is to turn a partial function into a total function know how this relates to programming Review the definitions of these functions: floor, ceiling (and the notations used for them in math notation) mod (and the notation used) gcd (greatest common divisor) log Review what composition of functions is. Review the terms: 1-to-1, injective, injection onto, surjective, surjection bijective, bijection

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