MATH 135 Lecture VIII Notes
Winter 2009
Binomial Theorem Recall that we are trying to come up with a way of expanding (a + b)n without actually having to expand it for each value of n in which we are interested. This is similar to wanting to come up with
MATH 135 Lecture VII Notes
Winter 2009
Binomial Coecients We want to determine a general expression for (a + b)n where n is a positive integer. This would save us from expanding this product every time. (Does this concept sound familiar?) The expansions f
MATH 135 Lectures V/VI/VII Notes
Winter 2009
Strong Induction Sometimes induction doesnt work where it looks like it should. We then need to change our approach a bit. The following example is similar to examples that weve done earlier. Lets try to make r
MATH 135 Lectures IV/V Notes
Winter 2009
Mathematical Induction The second technique of proof at which well look is mathematical induction. This is a technique that is normally used to prove that a statement is true for all positive integers n. Motivation
MATH 135 Lectures IV/V Notes
Winter 2009
Mathematical Induction The second technique of proof at which well look is mathematical induction. This is a technique that is normally used to prove that a statement is true for all positive integers n. Motivation
MATH 135 Lecture III Notes
Notation Some notation to remember: Z = cfw_. . . 3, 2, 1, 0, 1, 2, 3, . . . = set of integers P = cfw_1, 2, 3, 4, . . . = set of positive integers N.B. This notation is not standard, but is used by our textbook. R = the set of
MATH 135 Lecture II Notes
Winter 2009
Patterns and Conjectures We often use patterns in mathematics to guess what is happening in general (that is, make a conjecture), and then try to prove our pattern/conjecture (that is, turn it into a theorem). Conside
MATH 135 Lecture I Notes
Winter 2009
Introduction Two of the main purposes of MATH 135 are to teach you about proof (what it means, how to prove things, etc.) and to teach you about precision in mathematics. What is a Proof? A proof is A rigorous mathemat
Math 135
Assignment 2 Solutions
Winter 2009
1. Express each statement as a logical expression using quantiers. State the universe of discourse. (a) There is a smallest positive integer. ab 0 < a and a b, UD is the integers.
(b) There is no smallest positi
Math 135
1.
Assignment 6 Solutions
Winter 2009
a) Prove that 5n7 + 14n4 19n 0(mod 7) for all integers n. Solution Since 7 is prime, by Fermats Little Theorem, n7 n(mod 7). Thus 5n7 19n 5n 19n 14n 0(mod 7). Therefore 5n7 + 14n4 19n 14n4 + 0 0(mod 7), as de
Math 135
Assignment 7 Solutions
Winter 2009
1. (a) Using the Extended Euclidean Algorithm, we can nd the gcd(1653,2000) and the required integers x and y such that 1653x + 2000y = 77. x column 0 1 1 5 6 23 98 317 y column 1 0 1 4 5 19 81 262 r column 2000
MATH 135 Lecture IX Notes
Winter 2009
If and only if In mathematics, we often see statements of the form A if and only if B (A B ). (See Assignment 1.) This means (If A then B ) and (If B then A). The parentheses are here for mathematical reasons, not Eng