ITC 3106

• Number Groups
  • Real Numbers $\mathbb{R}$
  • Rings

Number Groups

$\mathbb{N} = \{0,1,2,...\}$ [natural numbers]

$\mathbb{Z} = \{m-n \mid m,n \in \mathbb{N}\}$ [integers]

$\mathbb{Q} = \{m/n \mid m,n \in \mathbb{Z}, n \neq 0\}$ [rational numbers]

$\mathbb{R}$ [real numbers]

$\mathbb{C}$ [complex numbers]

for which
$$\mathbb{N \subset Z \subset Q \subset R \subset C}$$

Real Numbers $\mathbb{R}$

Algebraic Properties:

• Addition: for each $a$ and $b$ there exists a real number $a+b$
• Multiplication: for each $a$ and $b$ there exists a real number $a \cdot b$
• Addition and Multiplication Together: for all $a,b,c \in \mathbb{R}$, we have the law $a \cdot (b+c) = a \cdot b + a \cdot c$
• Avoiding collapse: we assume $0 \neq 1$

Order Properties:

$\mathbb{R}$ has an order relation < to classify each number as positive, negative, or zero, based on their relation to 0. These properties interact with $+$ and $\cdot$ to manipulate inequalities.

Completeness Axiom:

The implication that there are no missing points in the real number line, unlike the rational set which excludes instances of irrational numbers.

Operational Axioms:

1. Only universal quantifiers, for all:
   $$(\forall a \in \mathbb{R})(\forall b \in \mathbb{R}) \space a+b=b+a$$
2. Contain an existential quantifier, exists, to prove the existence of something:
   $$(\exists 0 \in \mathbb{R})(\forall a \in \mathbb{R}) \space a+0=0+a=a$$

Rings

A structure $(R,+,\cdot)$ is a ring if $R$ is a non-empty set and $+$ and $\cdot$ and binary operations:
$$+: R \times R \to R, \quad (a,b) \mapsto a + b$$
$$\cdot: R \times R \to R, \quad (a,b) \mapsto a \cdot b$$
such that:

Addition:

1. associativity: $(\forall a,b,c \in R) \quad a+(b+c)=(a+b)+c$
2. zero element: $(\exists0 \in R)(\forall a \in R) \quad a+0=0+a=a$
3. inverses: $(\forall a \in R)(\exists -a \in R) \quad a+(-a)=(-a)+a=0$
4. commutativity: $(\forall a,b \in R) \quad a+b=b+a$

Multiplication:

1. associativity: $(\forall a,b,c \in R) \quad a \cdot (b \cdot c) = (a \cdot b) \cdot c$

Addition and Multiplication:

1. $(\forall a,b,c \in R) \quad a \cdot (b+c) = a \cdot b + a \cdot c \quad$ and $(a+b) \cdot c = a \cdot b + b \cdot c$

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Implies Operation $p \implies q$

$p$	$q$	$p \implies q$
F	F	T
F	T	T
T	F	F
T	T	T

False can imply true, but true will never imply false.