Chapter 5: Functions

1. Chapter 5: Functions Discrete Mathematical Structures: Theory and Applications

2. Learning Objectives • Learn about functions • Explore various properties of functions • Learn about sequences and strings • Become familiar with the representation of strings in computer memory • Learn about binary operations Discrete Mathematical Structures: Theory and Applications

3. Functions Discrete Mathematical Structures: Theory and Applications

4. Functions Discrete Mathematical Structures: Theory and Applications

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10. Functions • Every function is a relation • Therefore, functions on finite sets can be described by arrow diagrams. In the case of functions, the arrow diagram may be drawn slightly differently. • If f : A → B is a function from a finite set A into a finite set B, then in the arrow diagram, the elements of A are enclosed in ellipses rather than individual boxes. Discrete Mathematical Structures: Theory and Applications

11. Functions • To determine from its arrow diagram whether a relation f from a set A into a set B is a function, two things are checked: • Check to see if there is an arrow from each element of A to an element of B • This would ensure that the domain of f is the set A, i.e., D(f) = A • Check to see that there is only one arrow from each element of A to an element of B • This would ensure that f is well defined Discrete Mathematical Structures: Theory and Applications

12. Functions • Let A = {1,2,3,4} and B = {a, b, c , d} be sets • The arrow diagram in Figure 5.6 represents the relation f from A into B • Every element of A has some image in B • An element of A is related to only one element of B; i.e., for each a ∈ A there exists a unique element b ∈ B such that f (a) = b Discrete Mathematical Structures: Theory and Applications

13. Functions • Therefore, f is a function from A into B • The image of f is the set Im(f) = {a, b, d} • There is an arrow originating from each element of A to an element of B • D(f) = A • There is only one arrow from each element of A to an element of B • f is well defined Discrete Mathematical Structures: Theory and Applications

14. Functions • The arrow diagram in Figure 5.7 represents the relation g from A into B • Every element of A has some image in B • D(g ) = A • For each a ∈ A, there exists a unique element b ∈ B such that g(a) = b • g is a function from A into B Discrete Mathematical Structures: Theory and Applications

15. Functions • The image of g is Im(g) = {a, b, c , d} = B • There is only one arrow from each element of A to an element of B • g is well defined Discrete Mathematical Structures: Theory and Applications

16. Functions • Let h be the relation described by the arrow diagram in Figure 5.8 • Every element of A has some image in B; i.e., there is an arrow originating from each element of A to an element of B. Therefore, • D(h) = A • However,element 1 has two images in B; i.e., there are two arrows originating from 1, one going to a and another going to b, so h is not well defined. Thus, the first condition of Definition 5.1.1 is satisfied,but the second one is not. • Therefore, h is not a function Discrete Mathematical Structures: Theory and Applications

17. Functions • The arrow diagram in Figure 5.9 represents a relation from A into B • Not every element of A has an image in B. For example, the element 4 has no image in B. In other words, there is no arrow originating from 4 • Therefore, 4 D(k), so D(k)  A • This implies that k is not a function from A into B Discrete Mathematical Structures: Theory and Applications

18. Functions • Numeric Functions • If the domain and the range of a function are numbers, then the function is typically defined by means of an algebraic formula • Such functions are called numeric functions • Numeric functions can also be defined in such a way so that different expressions are used to find the image of an element Discrete Mathematical Structures: Theory and Applications

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21. Functions Example 5.1.16 • Let A = {1,2,3,4} and B = {a, b, c , d}. Let f : A → B be a function such that the arrow diagram of f is as shown in Figure 5.10 • The arrows from a distinct element of A go to a distinct element of B. That is, every element of B has at most one arrow coming to it. • If a1, a2∈ A and a1= a2, then f(a1) = f(a2). Hence, f is one-one. • Each element of B has an arrow coming to it. That is, each element of B has a preimage. • Im(f) = B. Hence, f is onto B. It also follows that f is a one-to-one correspondence. Discrete Mathematical Structures: Theory and Applications

22. Functions Example 5.1.18 • Let A = {1,2,3,4} and B = {a, b, c , d, e} • f : 1 → a, 2 → a, 3 → a, 4 → a • For this function the images of distinct elements of the domain are not distinct. For example 1  2, but f(1) = a = f(2). • Im(f) = {a} B. Hence, f is neither one-one nor onto B. Discrete Mathematical Structures: Theory and Applications

23. Functions • Let A = {1,2,3,4} and B = {a, b, c , d, e} • f : 1 → a, 2 → b, 3 → d, 4 → e • For this function, the images of distinct elements of the domain are distinct. Thus, f is one-one. In this function, for the element c of B, the codomain, there is no element x in the domain such that f(x) = c ; i.e., c has no preimage. Hence, f is not onto B. Discrete Mathematical Structures: Theory and Applications

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25. Functions • Let A = {1,2,3,4}, B = {a, b, c , d, e},and C = {7,8,9}. Consider the functions f : A → B, g : B → C as defined by the arrow diagrams in Figure 5.14. • The arrow diagram in Figure 5.15 describes the function h = g ◦ f : A → C. Discrete Mathematical Structures: Theory and Applications

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28. Special Functions and Cardinality of a Set Discrete Mathematical Structures: Theory and Applications

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46. Sequences and Strings Discrete Mathematical Structures: Theory and Applications

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50. m is the lower limit of the sum, n the upper limit of the sum, and ai the general term of the sum Discrete Mathematical Structures: Theory and Applications