1. 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

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

3. MON 9/9 FUNCTIONS (1.6)

4. 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

5. Examples
Exercise : 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 : 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.

6. Examples II
Exercise : Find the inverse function of f(x) = x3 + 1.
Exercise : 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.

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

8. 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

9. Examples
Exercise : 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 : Find a formula for k=0m k, when m is a positive integer. (Hint: use the formula for k=1n k3.)

10. Examples II
Exercise : 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

11. FRI 9/13 THE GROWTH OF FUNCTIONS (1.8)

12. 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)

13. Examples
Exercise 1.8.7(a): Find the least integer n such that f(x) = 2 x3 + x2 log x is O(xn).
Exercise : Show that f(x) is (g(x)) iff f(x) is O(g(x)) and g(x) is O(f(x)).
Exercise : 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.