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Definition of Axiom
Statement or proposition, accepted as true without proof.
Definition of Set
Collection of things labeled S, obtains elements/members
Binary Operation
Action on a set, they define a function. Typically depend on two or more elements.
Everything is connected to ______ and ______ in a set
addition, multiplication
Axiom 1.1 → Commutativity, associatory and distributivity
a+b = b+a (C)
(a+b)+c = a+(b+c) (A)
a(b+c) = ab + ac (D)
ab = ba (C)
(ab)c = a(bc) (A)
Axiom 1.2 → Addition Identity
There exists an integer 0 s.t a+0 = afor all integers a.
Axiom 1.3 → Multiplicative
Identity There exists an integer 1 such that a×1 = a for all integers a.
Axiom 1.4 → Additive Inverse
For every integer a, there exists an integer -a such that a + (-a) = 0.
Axiom 1.5 → Cancellation
If a, b, and c are integers and a + b = a + c, then b = c.
Properties of =
a = a (Reflexivity)
a=b, then b=a (symmetry)
a=b, b=c, then a=c (Transitivity)
a=b, then a can be replaced by b in any statement or expression
Properties of Does not equal
NOT reflexive
Symmetric
NOT transitive
How do you mark the end of a proof
with a square
Proposition 1.12 → (Uniqueness of the additive identity)
Suppose a ∈ Z. If a has the property that b + a = b for all b ∈ Z, then a = 0. In other words, the additive identity is unique.
What does ‘or’ mean in maths
In mathematics, 'or' refers to a logical disjunction where at least one of the statements must be true. It is often used in conditional statements and set theory to denote options or alternatives.
Definition 1.27 (Subtraction)
For a,b ∈ Z, we define a−b to be a+(−b). This new binary operation − will be called subtraction
Axiom 2.1 (Natural numbers)
There exists a subset N of Z with the following properties:
(i) If a,b ∈ N, then a+b ∈ N. (The subset N is closed under addition.)
(ii) If a,b ∈ N, then ab ∈ N. (The subset N is closed under multiplication.)
(iii) 0 ̸∈ N.
(iv) For every a ∈ Z, we have a ∈ N or a = 0 or −a ∈ N.
What is a natural number
A positive integer
What is a proof by contradiction
A method of mathematical proof where the negation of the statement to be proved is assumed, leading to a contradiction. This implies that the original statement must be true.
⊄
Not a subset
⊆
subset or equal subset
⊈
not a subset and not an equal set
⊊
strict subset
Set Notation
{nEZ : some property of n}
Axiom 2.17 (Induction Axiom).
Suppose a subset A ⊆ Z satisfies the following properties:
(i) 1 ∈ A
(ii) n ∈ A =⇒ n+1∈A.
Then N ⊆A
Theorem 2.19 (Principle of mathematical induction: first form).
Suppose that, for each k ∈N, we have a statement P(k). Furthermore, suppose that
(i) P(1) is true
(ii) for all n ∈ N, P(n) ⇒ P(n+1).
Then P(k) is true for all k ∈ N
What are the steps of induction
Identify p(x)
Prove base case (p(1))
prove p(x+1), Remember to use induction hypothesis
Theorem 2.26 (Principle of mathematical induction: first form revisited)
Suppose m is a fixed integer and that, for each k ∈ Z with k ≥ m, we have a statement P(k). Furthermore, suppose that (i) P(m) is true, and (ii) for all n ≥ m, P(n) =⇒ P(n+1). Then P(k) is true for all k ≥ m
Definition 2.28 (Smallest and greatest elements).
Suppose A ⊆ Z is nonempty. If there exists m ∈ A such that m≤afor all a ∈ A, then we say m is a smallest element of A and write m = min(A). If there exists M ∈ A such that M ≥afor all a ∈ A, then we say M is a greatest element of A and write M = max(A)
Theorem 2.33 (Well-ordering principle).
Every nonempty subset of N has a smallest element.
Definition 2.36 (gcd).
Suppose a,b ∈ Z. If a and b are not both zero, we define gcd(a,b) = min {k ∈ N : k = ax+by for some x,y ∈ Z} .
∀
for all
∃
there exists
(∃0 ∈ Z such that)(∀a ∈ Z)
Quantified Statement
∄
Does not exist
∃!
There exists a unique
⇒
Implies
⇐⇒
If and only if
What is the converse
Reversing the Statements, for example p implies q becomes q implies p.
What is the contrapositive
Negating the expressions and reversing, creates equivalent statements for example P implies Q becomes not Q → not P
what is the negation of ‘and’
or
What is the negation of ‘or’
and
¬
negation (or not)
What is De Morgans laws
¬(A ∧ B) is equivalent to (¬A ∨ ¬B), and ¬(A ∨ B) is equivalent to (¬A ∧ ¬B).
∧
and
∨
or
How to negate quantifiers
Same place, swap symbol (for all → exists)
Negate final statement