Chapter 1 Marlow Anderson Flashcards
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What is the set of natural numbers (N)?
The set of counting numbers:
N = {1, 2, 3, 4, …}
It is closed under addition.
What is the set of integers (Z)?
The set:
Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
The smallest set containing N that is closed under subtraction.
How is multiplication defined in N?
As repeated addition:
n * a = a + a + … + a (n times)
What is the Well-ordering Principle?
Every non-empty subset of N has a least element.
State the Principle of Mathematical Induction.
Suppose X ⊆ N satisfies:
1. 1 ∈ X
2. If k ∈ X for all k < n, then n ∈ X
Then X = N.
What is the Strong Principle of Mathematical Induction?
Suppose X ⊆ N satisfies:
1. 1 ∈ X
2. If n > 1 and n - 1 ∈ X, then n ∈ X
Then X = N.
Define the Fibonacci sequence.
a₁ = 1, a₂ = 1, aₙ₊₂ = aₙ₊₁ + aₙ for n ≥ 1
What inequality holds for Fibonacci numbers?
aₙ₊₁ ≤ 2aₙ for all n ≥ 1
How many subsets does a set with n elements have?
Exactly 2ⁿ subsets.
What is the sum of the first n odd positive integers?
1 + 3 + 5 + … + (2n - 1) = n²
What is Theorem 1.1?
The Well-ordering Principle implies the Principle of Mathematical Induction.
What is the axiomatic method in mathematics?
A method where mathematics is built from a minimal set of assumptions (axioms), and all other results are logically deduced from them.
