Overview

CADI Properties of Addition

» Closure Property

→ if $x,y\in \mathbb{R}$, then $x+y\in \mathbb{R}$

» Commutative Property

→ $x+y=y+x$

» Associative Property

→ $(x+y)+z=x+(y+z)$

» Additive Identity Property

→ $0\in \mathbb{R}$, such that $x+0=x$

» Additive Inverse Property

→ $-x\in \mathbb{R}$ for any $x\in \mathbb{R}$ such that $x+(-x)=0$

» Subtraction is to be handled as additive inverse for the properties

*this is important as algebra extensively involves these properties*

→ Commutative property involving subtraction : $x-y$ is given as $x+(-y)=(-y)+x$

→ Associative property involving subtraction : $(x-y)-z$ is given as $(x+(-y))+(-z)=x+((-y)+(-z))$

closed means within

Consider the numbers $2$ and $3$. Is $2+3$ a real number? "yes, a real number".

That is, for any real numbers $p,q$, $p+q$ is always a real number

"closure" means closed and not open

Closure Property of Addition: Given $p,q\in \mathbb{R}$, $p+q\in \mathbb{R}$.

Closure Property applied to Subtraction: Given $p,q\in \mathbb{R}$. $p-q\in \mathbb{R}$.

Proof:

Given $p,q\in \mathbb{R}$

$-q\in \mathbb{R}$ *as per Additive Inverse Property*

$p+(-q)\in \mathbb{R}$ *as per Closure property of Addition*

$\Rightarrow p-q\in \mathbb{R}$

Using Closure Property: Given $p,q,r,s\in \mathbb{R}$, $p+q-r+s$, the subexpression $p+q-r$ is a real number and can be considered as a single number for any other property.

For example as per commutative property $p+q-r+s=s+p+q-r$, in which $p+q-r$ is considered to be a single real number.

Consider the numbers $2$ and $3$. Which of the following is true?

$2+3=3+2$ or

$3+2$ does not equal to $2+3$

The answer is "$2+3=3+2$".

Given $p,q\in \mathbb{R}$; $p+q=q+p$

forward and backward

The word "commute" means "to go to and fro between two places on a regular basis".

Commutative Property of Addition: Given $p,q\in \mathbb{R}$, $p+q=q+p$.

Commutative Property applied to Subtraction: $p-q=-q+p$.

Note: Subtraction has to be handled as inverse of addition, $p-q=p+(-q)$ and then commutative property can be used.

Using Commutative Property: Given $p,q,r\in \mathbb{R}$, the expression $p+q+p-q+r-p+2r$ is simplified to $p+3r$. *students may work this out to understand.*

with this or that

Given $p,q,r\in \mathbb{R}$. The expression $(p+q)+r$ equals $p+(q+r)$

In the first expression, $q$ is added first with $p$ and then $r$ is added to the result.

In the second expression $q$ is added first with $r$ and then $p$ is added to the result.

Either way, the result is same.

For example, $(2+3)+7=2+(3+7)$

The word "associate" means 'to connect with; to join'.

Associative Property of Addition: Given $p,q,r\in \mathbb{R}$. $(p+q)+r=p+(q+r)$.

Associative Property applied to subtraction: $(p-q)-r=p+(-q-r)$.

Note: Subtraction has to be handled as inverse of addition, $(p-q)-r=(p+(-q))+(-r)$ and then associative property can be used.

zero

Given $p\in \mathbb{R}$. What is $p+0=p$

Additive Identity Property: For any $p\in \mathbb{R}$, there exists $0\in \mathbb{R}$ such that $p+0=p$.
Additive Identity applied to Subtraction: $p-0=p$

Note: $-0=0$ and so $p-0=p+(-0)=p+0=p$.

inverse

Given $p\in \mathbb{R}$. What is $p-p=0$

Additive Inverse Property: For any $p\in \mathbb{R}$, there exists $-p\in \mathbb{R}$ such that $p+(-p)=0$.

summary

The properties together are named as **CADI** properties of addition. The abbreviation CADI is a simplified form of the first letters of Closure, Commutative, Associative, Distributive, Inverse, and Identity properties.

*Note: Distributive property is shared with multiplication and is explained in the next page. *

**LPA: CADI properties of Addition **

• Closure Property

if $x,y\in \mathbb{R}$, then $x+y\in \mathbb{R}$

• Commutative Property

$x+y=y+x$

• Associative Property

$(x+y)+z=x+(y+z)$

• Additive Identity Property

$0\in \mathbb{R}$, such that $x+0=x$

• Additive Inverse Property

$-x\in \mathbb{R}$ for any $x\in \mathbb{R}$ such that $x+(-x)=0$

Subtraction is to be handled as additive inverse and properties of addition applies to subtraction in the form of addition.

This is important as algebra extensively uses these properties.

→ *Commutative property involving subtraction* : $x-y$ is given as $x+(-y)$$=(-y)+x$

→ * Associative property involving subtraction* : $(x-y)-z$ is given as $(x+(-y))+(-z)$$=x+((-y)+(-z))$

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__