Week #3 – 9/11/13 September 2002 Prof. Marie desJardins CMSC 203 / 0201 Fall 2002 Week #3 – 9/11/13 September 2002 Prof. Marie desJardins

TOPICS Functions Sequences and summations Growth of functions; big-O notation

MON 9/9 FUNCTIONS (1.6)

CONCEPTS / VOCABULARY Function/mapping Domain, codomain, image, pre-image One-to-one/injective, onto/surjective, one-to-one correspondence/bijective Inverse/invertible functions, compositions, graphs Floor, ceiling

Examples Exercise 1.6.15: Determine whether each of the following functions is a bijection from R to R. (a) f(x) = 2x + 1 (b) f(x) = x2 + 1 (c) f(x) = x3 (d) f(x) = (x2 + 1) / (x2 + 2) Exercise 1.6.33: Show that x - ½ is the closest integer to the [real] x, except when x is midway between two integers, when it is the smaller of these two integers.

Examples II Exercise 1.6.55: Find the inverse function of f(x) = x3 + 1. Exercise 1.6.58: Suppose that f is a function from A to B, where A and B are finite sets with |A| = |B|. Show that f is one-to-one if and only if it is onto.

WED 9/11 SEQUENCES AND SUMMATIONS (1.7) ** Homework #1 due today! ** ** (Ungraded) quiz today! **

CONCEPTS / VOCABULARY Sequences, terms, strings Arithmetic progressions, geometric progressions, geometric series Summation , index of summation, lower and upper limit Standard summation formulas: Table 1.6.2 Cardinality, countability

Examples Exercise 1.7.19: Show that j=1n (aj – aj-1) = an – a0 where a0, a1, …, an is a sequence of real numbers. This type of sum is called telescoping. Exercise 1.7.25: Find a formula for k=0m k, when m is a positive integer. (Hint: use the formula for k=1n k3.)

Examples II Exercise 1.7.31: Determine whether each of the following sets is countable or uncountable. For those that are countable, exhibit a one-to-one correspondence between the set of natural numbers and that set. (a) the negative integers (b) the even integers (c) the real numbers between 0 and ½ (d) integers that are multiples of 7

FRI 9/13 THE GROWTH OF FUNCTIONS (1.8)

CONCEPTS / VOCABULARY Big-O notation (upper bound on growth of f(x)) f(x) is O(g(x)) if there exist constants C and k such that |f(x)|  C |g(x)| whenever x  k Triangle inequality Growth of (f1 + f2)(x), (f1 f2)(x) Big-Omega  (lower bound on growth of f(x)) Big-Theta  (upper and lower bound)

Examples Exercise 1.8.7(a): Find the least integer n such that f(x) = 2 x3 + x2 log x is O(xn). Exercise 1.8.25: Show that f(x) is (g(x)) iff f(x) is O(g(x)) and g(x) is O(f(x)). Exercise 1.8.39: If f1(x) and f2(x) are functions from the set of positive integers to the set of positive real numbers, and f1(x) and f2(x) are both (g(x)), is (f1- f2)(x) also (g(x))? Either prove that it is or give a counterexample.