Chapter2 - Chapter 2 Principle of Mathematical Induction...

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Unformatted text preview: Chapter 2 Principle of Mathematical Induction and Properties of Numbers We will introduce some basic material that will be used throughout the rest of the course. 2.0 Notation We will use the following notation: • N will denote the set of natural numbers { 1 , 2 , 3 ,... } . • Z will denote the set of integers { ...,- 2 ,- 1 , , 1 , 2 ,... } . • Q will denote the set of rational numbers a b : a ∈ Z ,b ∈ N . • R will denote the set of real numbers. Intervals. We will use the notation ( a,b ) to denote the set { x : a < x < b } . This is called an open interval . We will use [ a,b ] to denote the set { x : a ≤ x ≤ b } . This is called a closed interval . Additionally, we will use the nota- tion (-∞ ,b ), ( a, ∞ ), (-∞ ,b ], [ a, ∞ ) to mean the open intervals { x : x < b } , { x : x > a } , and the closed intervals { x : x ≤ b } , { x : x ≥ a } , respec- tively. Finally, we will use [ a,b ) and ( a,b ] to denote the half-open intervals { x : a ≤ x < b } and { x : a < x ≤ b } , respectively. Formally, we make the following definition. Definition 2.0.1. A set S ⊂ R is an interval if for every x , y ∈ S , if x ≤ z ≤ y then we must have z ∈ S . 10 It is easy to see that the singleton set { a } is an interval for any a ∈ R . It is somewhat less obvious that the empty set , denoted by ∅ , is also an interval. To see why this is so we first ask what would it mean if the empty set was “not an interval”. In this case, we would have to be able to find a pair x , y ∈ ∅ and an element z ∈ R such that x ≤ z ≤ y but z 6∈ ∅ . This is clearly impossible because no such x , y exist in ∅ . As such, we have shown that the statement, “ ∅ is not an interval” is false, and as such we have proved that ∅ is an interval. Definition 2.0.2. An interval I is said to be degenerate if I = { c } for some c ∈ R or if I = ∅ . Otherwise, we say that it is nondegenerate . We will use the notation A ⊂ B and A ⊆ B interchangeably to mean that A is a subset of B with the possibility that A = B though when we explicitly wish to emphasize that A = B is a possibility, we will generally use A ⊆ B . When we wish to express that A is a proper subset of B , then we can either specify further that A 6 = B , or we can use the notation A B . We will let B \ A = { x ∈ B | x 6∈ A } . In the special case when B = R , we call the set R \ A the complement of A in R and denote this set by A c . 2.1 Mathematical Induction Mathematics is built on axioms . Axioms are mathematical statements that we accept as being true without need for proof. The following axiom introduces one of the fundamental properties of the set N of natural numbers. It will lead to an important method of proof called “proof by induction,”....
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This note was uploaded on 10/21/2010 for the course MATH 147 taught by Professor Wolzcuk during the Fall '09 term at Waterloo.

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Chapter2 - Chapter 2 Principle of Mathematical Induction...

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