2.1 The Algebraic and Order Properties of R

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NOTE

Consider the conditional statement $p\to q$. Where $p$ is a hypothesis, $q$ is a conclusion.

• Negation: $\neg p$
• Converse: $q\to p$
• Inverse: $\neg p\to \neg q$
• Biconditional: $\neg p⟺\neg q$
• Contrapositive: $\neg q\to \neg p$

An important property of contrapositive is that: $(p\to q)⟺(\neg q\to \neg p)$

2.1.1 Definition: Field axioms

Definition

Property	Premise	Addition	Multiplication
Commutative	$\forall x,y\in \mathbb{R}$	$x+y=y+x$	$x\cdot y=y\cdot x$
Associative	$\forall x,y,z\in \mathbb{R}$	$(x+y)+z=x+(y+z)$	$(x\cdot y)\cdot z=x\cdot (y\cdot z)$
Identity Element	$\exists 0,1\in \mathbb{R},0\ne 1\ni \forall x\in \mathbb{R}$	$x+0=x=0+x$	$x\cdot 1=x=x\cdot 1$
Inverse Element	$\forall x,y\in \mathbb{R},y\ne 0\exists (-x),y^{-1}\in \mathbb{R}$	$x+(-x)=0=(-x)+x$	$y\cdot y^{-1}=1=y^{-1}\cdot y$
Distributive	$\forall x,y,z\in \mathbb{R}$	$x(y+z)=xy+xz=(y+z)x$

==All familiar techniques of algebra can be derived from those 9 properties.==

2.1.2 Theorem: Uniqueness of identity and multiplication by 0

Theorem

Let $a,b,u,z\in \mathbb{R}$

1. $(z+a=a)\Rightarrow z=0$
2. $(b\ne 0\wedge u\cdot b=b)\Rightarrow u=1$
3. $a\cdot 0=0$

2.1.3 Theorem: Uniqueness of reciprocal and multiplication resulting in 0

Theorem

Let $a,b\in \mathbb{R}$

1. $(a\ne 0\wedge a\cdot b=1)\Rightarrow b=1/a$
2. $a\cdot b=0\Rightarrow (a=0\vee b=0)$

Various notation in algebra

Let $a,b\in \mathbb{R},n\in \mathbb{N},m,n\in \mathbb{Z}$

$$\begin{array}{rl}ab& :=a\cdot b\\ a-b& :=a+(-b)\\ \frac{a}{b}& :=a\cdot \frac{1}{b}\\ a^n& :=a\cdot a\cdot a\cdot \cdots \cdot a\\ a^{-n}& :={\left(\frac{1}{a}\right)}^n\\ a^{m+n}& :=a^m{a}^n\end{array}$$

Definition: Natural numbers and integers

Definition

Natual numbers $(\mathbb{N})$: The elements $n\in \mathbb{N}$ are identified as $n-$fold sum of the unit element $1\in \mathbb{R}$.

Definition

Integers $(\mathbb{Z})$: The elements $z\in \mathbb{Z}$ are identified as

• $0\in \mathbb{R}$
• $n-$fold of $1\in \mathbb{R}$
• $n-$ fold of $-1\in \mathbb{R}$

NOTE

Notice that $\mathbb{N}$ and $\mathbb{Z}$ are subsets of $\mathbb{R}$

Definition: Rational and irrational numbers

Definition

Rational numbers $(\mathbb{Q})$: Elements in $\mathbb{Q}$ are those that can be written in the form of

$$Q:=\left\{\frac{p}{q}:p,q\in \mathbb{Z},q\ne 0\right\}$$

Definition

Irrational numbers $({\mathbb{Q}}^C)$: Defined as real numbers not in $\mathbb{Q}$:

$${\mathbb{Q}}^C=\mathbb{R}-\mathbb{Q}$$

The word "irrational" come from the fact that they cannot be expressed as "ratios" of integers.

2.1.4 Theorem: Square root of 2 is irrational

Theorem

There does not exist a rational number $r$ such that $r^2=2$.

2.1.5 Definition: The order properties of R

Definition

There exists a nonempty subset $\mathbb{P}$ of $\mathbb{R}$, called the set of positive real numbers, that satisfies the following properties:

1. $a,b\in \mathbb{P}⟹a+b\in \mathbb{P}$
2. $a,b\in \mathbb{P}⟹ab\in \mathbb{P}$
3. If $a\in \mathbb{R}$, then exactly one of the following holds: $$ a \in \mathbb P, \qquad a = 0, \qquad -a \in \mathbb P $

The first two properties ensure compatibility with addition and multiplication operations.

The last one is often called the tricothomy property

• If $a\in \mathbb{P}$, we write $a>0$ and say that $a$ is a real positive number.
• If $a\in \mathbb{P}\cup \{0\}$, we write $a\ge 0$ and say that $a$ is a real nonnegative number.
• If $-a\in \mathbb{P}$, we write $a<0$ and say that $a$ is a real negative number.
• If $-a\in \mathbb{P}\cup \{0\}$, we write $a\le 0$ and say that $a$ is a real nonpositive number.

2.1.6 Definition: Some inequality notations

Definition

Let $a,b\in \mathbb{R}$.

• If $a-b\in \mathbb{P}$, then we write $a>b$ or $b<a$
• If $a-b\in \mathbb{P}\cup \{0\}$, then we write $a\ge b$ or $b\le a$

Based on the Tricothomy Property, if both $a\le b$ and $a\ge b$, then $a=b$.

For notational convenience, we can write $a<b$ and $b<c$ as $a<b<c$. Also applies to the “or equal to” notation.

2.1.7 Theorem: Inference from inequalities

Theorem

If $ab>0$, then either

1. $a>0$ and $b>0$, or
2. $a<0$ and $b<0$.

2.1.8 Theorem: Various order theorems

Theorem

1. If $a\in \mathbb{R}$ and $a\ne 0$, then $a^2>0$.
2. $1>0$.
3. If $n\in \mathbb{N}$, then $n>0$.

Proposition: No “smallest” positive real number can exist

Proposition

Notice that if $a\in \mathbb{P}$, and since $\frac{1}{2}>0$, we have

$$0<\frac{1}{2}a<a$$

Thus if it is claimed that $a$ is the smallest real positive number, we can obtain a smaller positive number $\frac{1}{2}a$.

2.1.9 Theorem: Notion of limit from order

Theorem

If $a\in \mathbb{R}$ is such that $0\le a<\epsilon$ for every $\epsilon >0$, then $a=0$.

2.1.10 Theorem: Inference from multiplication and inequality

Theorem

If $ab>0$, then either

1. $a>0$ and $b>0$, or
2. $a<0$ and $b<0$.

2.1.11 Corollary: From 2.1.10

Corollary

If $ab<0$, then either

1. $a<0$ and $b>0$, or
2. $a>0$ and $b<0$.