Name:
CAMOSUN
g.OTtO)0-g
MATH 126 (Winter, 2014)
Term Test 3
by George Ballinger
Answer the questions in the space provided.
This test has 10 questions for a total of 25 marks.
Express answers to counting problems as integers and show your calculations.
1
Classic Open Problems in Number Theory
as of January 2014
Largest Known Prime Number: Can you find a prime number greater than all known
primes?
There are infinitely many prime numbers. As such, there is no largest prime number.
The probability that a lar
Big-O Example
We show that log n! is O(n log n) and n log n is O(log n!).
We use the following well-known properties of logarithms.
log(ab) = log a + log b, the log of a product equals the sum of the logs
a b ! log a log b, the log function is increasing
Arguments
Example 1
If you are Vulcan, then you reason logically.
You are Vulcan.
) You reason logically.
Example 2
If the water is warm, then I will go swimming.
The water is not warm.
) I will not go swimming.
Example 3
Hawaii is an island if the sky is
Comparing the Growth of Functions
The following functions of n (where n is a positive integer) are listed in order from
slowest
to fastest growing. Each function in this list is big-O of the functions below it. For ease of
comparison these functions have
Conditional Statements
Definition If p and q are propositions, then p ! q is called a conditional statement (or
an implication) and is read if p, then q. The statement is true if both p and q are true, if
p is false and q is true, or if both p and q are f
go LUT/o N5
CAMOSUN Name:
MATH 126 (Winter, 2014)
Term Test 1
by George Ballinger
Answer the questions in the space provided.
This test has 16 questions for a total of 25 marks.
1. (2 marks) Construct a truth table for the proposition (p ) q) A (p V
4'
CAMOSUN
Name:
M A T H 126 (Winter,
Term Test 2
SouUTiOfiSS
2014)
by George Ballinger
Answer the questions in the space provided.
This test has 12 questions for a total of 25 marks.
1. (2 marks) Consider the factorial function / : N N given by f(n) n\e
Example of a Proof Using Logical Equivalences
Use Logical Equivalences to prove that [(p (p q) (p q)] p is a tautology.
Proof:
[(p (p q) (p q)] p [(p (p) q) (p q)] p De Morgans law
[(p (p q) (p q)] p Double Negation law
[(p p) q) (p q)] p Associative la