Functions and Sequences: Concepts and Examples, Assignments of Discrete Mathematics

Download Functions and Sequences: Concepts and Examples and more Assignments Discrete Mathematics in PDF only on Docsity! Reading Assignment, Due Sep 17 1. A function f : A → B describes a particular kind of relationship between two sets A and B; it maps a unique element of B to every element of A. As was the case for sets, you must be already familiar with functions. There are some concepts, however, that may be new. E.g., injective, surjective, and bijective functions; and the composition of functions. Please read the entire §2.3 “Functions.” 2. Recall that sets do not care about the order of elements. When order is important, we use a discrete structure called a sequence. Read §2.4 “Sequences and Summations” till end of Example 17 on page 158. A lot of this should also be already familiar. Two kinds of sequences of numbers are very common: arithmetic pro- gressions and geometric progressions. The subsection on “Special Integer Sequences” is interesting, and you must have seen puzzles such as in Ex- amples 5 – 8 earlier; but there will not be any such puzzles asked in this course. Table 2 (pg 157) has a list of summation formulae. E.g., the sum of the first n terms of the form k3 is n2(n + 1)2/4. We will later learn of an elegant tool called Induction that will help us prove that this is really so. 3. Turn in answers to the following: (a) Give an interesting example of a function which is — i. Injective but not surjective. ii. Surjective but not injective. iii. Both injective and surjective. iv. Neither injective nor surjective. My answer is a function that — i. Maps each song on an ipod to the memory address where the song’s first byte is stored. Note that all memory addresses don’t necessarily contain the first byte of some song. ii. Maps each student to the major they are enrolled in. (Assume no one is double-majoring, and every major has some enrollment.) iii. Maps each person to their finger prints. iv. Maps each student to their favourite song. (b) Definition 1 on page 150 defines a sequence as a function from (a subset of) the set of integers to some set of interest S. What is the intuitive reason behind doing this? Could we instead say a sequence is a function from the set of interest S to the set of integers? 1