Overview

Comparison

» Trichotomy Property

*"tri-" means three and "-chotomy" means division-into-parts.*

→ Given $a,b\in \mathbb{R}$ one and only one of the following is true.

$a<b$

$a=b$

$a>b$

» Transitivity

*"trans-" means across and "-itivity" means to-go or to-pass*

→ If $a=b$ and $b=c$, then it implies $a=c$

→ If $a<b$ and $b<c$, then it implies $a<c$

→ If $a>b$ and $b>c$, then it implies $a>c$

three possibilities

Given two real numbers $p$ and $q$, which of the following is correct?

Trichotomy Property: For any numbers $p,q\in \mathbb{R}$, *one and only one * of the following is true.

$p<q$

$p=q$

$p>q$

Note: sometimes, $p\le q$ is considered. This specifies, consider pair of numbers $p$ and $q$ such that $p$ is less than or equal to $q$.

It specifies which numbers to choose. That is, if $q$ is 4, then $p$ can be any negative number or $0,1,2,3,4$. It defines a set of numbers for $p$.

In trichotomy property, for any two numbers (for example, $4$ and $3$), one and only one of the three possibilities is true.

The word "trichotomy" means: Division into three parts.

tri- means three and -chotomy means division-into-parts

pass across

For three real numbers $p,q,r$, it is given that $p<q$ and $q<r$, then $p<r$

Transitivity Property: For three numbers $p,q,r\in \mathbb{R}$,

• if $p<q$ and $q<r$, then $p<r$.

• if $p=q$ and $q=r$, then $p=r$.

• if $p>q$ and $q>r$, then $p>r$.

Note 1: If $p<q$ and $q>r$, then the relation between $p$ and $r$ cannot be ascertained.

Note 2: If $p<q$ and $q=r$, then $q$ can be replaced by $r$ to get $p<r$.

The word "transitivity" means: some property passes across from one to another.

trans- means across and -itivity means to-go or to-pass.

summary

**LPA - Comparison** : Comparison of numbers has Trichotomy and Transitivity properties.

» **Trichotomy Property**

*"tri-" means three and "-chotomy" means division-into-parts.*

→ Given $a,b\in \mathbb{R}$ one and only one of the following is true.

$a<b$

$a=b$

$a>b$

» **Transitivity Property**

*"trans-" means across and "-itivity" means to-go or to-pass*

→ If $a=b$ and $b=c$, then it implies $a=c$

→ If $a<b$ and $b<c$, then it implies $a<c$

→ If $a>b$ and $b>c$, then it implies $a>c$

Outline

The outline of material to learn "Algebra Foundation" is as follows.

Note: *click here for detailed outline of Foundation of Algebra*

→ __Numerical Arithmetics__

→ __Arithmetic Operations and Precedence__

→ __Properties of Comparison__

→ __Properties of Addition__

→ __Properties of Multiplication__

→ __Properties of Exponents__

→ __Algebraic Expressions__

→ __Algebraic Equations__

→ __Algebraic Identities__

→ __Algebraic Inequations__

→ __Brief about Algebra__