CHAPTER 7 FUNCTIONS Copyright © Cengage Learning. All rights reserved. SECTION 7.1 Functions Defined on General Sets Copyright © Cengage Learning. All rights reserved. Functions Defined on General Sets We have already defined a function as a certain type of relation. The following is a restatement of the definition of function that includes additional terminology associated with the concept. 3 Arrow Diagrams 4 Arrow Diagrams We have known that if X and Y are finite sets, you can define a function f from X to Y by drawing an arrow diagram. You make a list of elements in X and a list of elements in Y, and draw an arrow from each element in X to the corresponding element in Y, as shown in Figure 7.1.1. Figure 7.1.1 5 Arrow Diagrams This arrow diagram does define a function because 1. Every element of X has an arrow coming out of it. 2. No element of X has two arrows coming out of it that point to two different elements of Y. 6 Example 2 – A Function Defined by an Arrow Diagram Let X = {a, b, c} and Y = {1, 2, 3, 4}. Define a function f from X to Y by the arrow diagram in Figure 7.1.3. a. Write the domain and co-domain of f. b. Find f(a), f(b), and f(c). c. What is the range of f? d. Is c an inverse image of 2? Is b an inverse image of 3? e. Find the inverse images of 2, 4, and 1. f . Represent f as a set of ordered pairs. Figure 7.1.3 7 Example 2 – Solution a. domain of co-domain of b. c. range of d. Yes, No e. inverse image of inverse image of inverse image of (since no arrows point to 1) f. 8 Arrow Diagrams In Example 2 there are no arrows pointing to the 1 or the 3. This illustrates the fact that although each element of the domain of a function must have an arrow pointing out from it, there can be elements of the co-domain to which no arrows point. Note also that there are two arrows pointing to the 2—one coming from a and the other from c. 9 Arrow Diagrams Earlier we have given a test for determining whether two functions with the same domain and co-domain are equal, saying that the test results from the definition of a function as a binary relation. We formalize this justification in Theorem 7.1.1. 10 Example 3 – Equality of Functions a. Let J3 = {0, 1, 2}, and define functions f and g from J3 to J3 as follows: For all x in J3, Does f = g? b. Let F: R → R and G: R → R be functions. Define new functions F + G: R → R and G + F: R → R as follows: For all x R, Does F + G = G + F? 11 Example 3 – Solution a. Yes, the table of values shows that f(x) = g(x) for all x in J 3. b. Again the answer is yes. For all real numbers x, Hence F + G = G + F. 12 Examples of Functions 13 Example 4 – The Identity Function on a Set Given a set X, define a function IX from X to X by for all x in X. The function IX is called the identity function on X because it sends each element of X to the element that is identical to it. Thus the identity function can be pictured as a machine that sends each piece of input directly to the output chute without changing it in any way. Let X be any set and suppose that elements of X. Find and and . are 14 Example 4 – Solution Whatever is input to the identity function comes out unchanged, so and 15 Examples of Functions 16 Example 8 – The Logarithmic Function with Base b Find the following: a. b. d. c. e. Solution: a. b. c. 17 Example 8 – Solution d. because the exponent to which 2 must be raised to obtain 2m is m. e. because log2 m is the exponent to which 2 must be raised to obtain m. cont’d 18 Examples of Functions We have known that if S is a nonempty, finite set of characters, then a string over S is a finite sequence of elements of S. The number of characters in a string is called the length of the string. The null string over S is the “string” with no characters. It is usually denoted and is said to have length 0. 19 Example 9 – Encoding and Decoding Functions Digital messages consist of finite sequences of 0’s and 1’s. When they are communicated across a transmission channel, they are frequently coded in special ways to reduce the possibility that they will be garbled by interfering noise in the transmission lines. For example, suppose a message consists of a sequence of 0’s and 1’s. A simple way to encode the message is to write each bit three times. Thus the message would be encoded as 20 Example 9 – Encoding and Decoding Functions cont’d The receiver of the message decodes it by replacing each section of three identical bits by the one bit to which all three are equal. Let A be the set of all strings of 0’s and 1’s, and let T be the set of all strings of 0’s and 1’s that consist of consecutive triples of identical bits. The encoding and decoding processes described above are actually functions from A to T and from T to A. 21 Example 9 – Encoding and Decoding Functions cont’d The encoding function E is the function from A to T defined as follows: For each string s A, E(s) = the string obtained from s by replacing each bit of s by the same bit written three times. The decoding function D is defined as follows: For each string t T, D(t) = the string obtained from t by replacing each consecutive triple of three identical bits of t by a single copy of that bit. 22 Example 9 – Encoding and Decoding Functions cont’d The advantage of this particular coding scheme is that it makes it possible to do a certain amount of error correction when interference in the transmission channels has introduced errors into the stream of bits. If the receiver of the coded message observes that one of the sections of three consecutive bits that should be identical does not consist of identical bits, then one bit differs from the other two. In this case, if errors are rare, it is likely that the single bit that is different is the one in error, and this bit is changed to agree with the other two before decoding. 23 Boolean Functions 24 Boolean Functions We have discussed earlier that how to find input/output tables for certain digital logic circuits. Any such input/output table defines a function in the following way: The elements in the input column can be regarded as ordered tuples of 0’s and 1’s; the set of all such ordered tuples is the domain of the function. The elements in the output column are all either 0 or 1; thus {0, 1} is taken to be the co-domain of the function. The relationship is that which sends each input element to the output element in the same row. 25 Boolean Functions 26 Example 11 – A Boolean Function Consider the three-place Boolean function defined from the set of all 3-tuples of 0’s and 1’s to {0, 1} as follows: For each triple (x1, x2, x3) of 0’s and 1’s, Describe f using an input/output table. Solution: 27 Example 11 – Solution cont’d The rest of the values of f can be calculated similarly to obtain the following table. 28 Checking Whether a Function Is Well Defined 29 Checking Whether a Function Is Well Defined It can sometimes happen that what appears to be a function defined by a rule is not really a function at all. To give an example, suppose we wrote, “Define a function f : R → R by specifying that for all real numbers x, There are two distinct reasons why this description does not define a function. For almost all values of x, either (1) there is no y that satisfies the given equation or (2) there are two different values of y that satisfy the equation. 30 Checking Whether a Function Is Well Defined For instance, when x = 2, there is no real number y such that 22 + y2 = 1, and when x = 0, both y = –1 and y = 1 satisfy the equation 02 + y2 = 1. In general, we say that a “function” is not well defined if it fails to satisfy at least one of the requirements for being a function. 31 Example 12 – A Function That Is Not Well Defined We know that Q represents the set of all rational numbers. Suppose you read that a function f : Q → Z is to be defined by the formula for all integers m and n with n 0. That is, the integer associated by f to the number Is f well defined? Why? is m. 32 Example 12 – Solution The function f is not well defined. The reason is that fractions have more than one representation as quotients of integers. For instance, Now if f were a function, then the definition of a function would imply that since 33 Example 12 – Solution cont’d But applying the formula for f, you find that and so This contradiction shows that f is not well defined and, therefore, is not a function. 34 Checking Whether a Function Is Well Defined Note that the phrase well-defined function is actually redundant; for a function to be well defined really means that it is worthy of being called a function. 35 Functions Acting on Sets 36 Functions Acting on Sets Given a function from a set X to a set Y, you can consider the set of images in Y of all the elements in a subset of X and the set of inverse images in X of all the elements in a subset of Y. 37 Example 13 – The Action of a Function on Subsets of a Set Let X = {1, 2, 3, 4} and Y = {a, b, c, d, e}, and define F : X → Y by the following arrow diagram: Let A = {1, 4}, C = {a, b}, and D = {c, e}. Find F(A), F(X), F−1(C), and F−1(D). 38 Example 13 – Solution 39