Formulas in Pure Maths

Formulas in Pure Maths (M1)

Module 1

Table of Contents

Prepositional Logic : Slide 3

Real Number System : Slide 7

Binary operations : Slide 9

Surds : Slide 10

Summation Notation : Slide 11

Polynomials : Slide 12

Modulus Function : Slide 14

Cubic Equations : Slide 16

Functions : Slide 18

Logs : Slide 19

Prepositional Logic (Module 1)

* Conjunction (and) :  $\wedge$  
* Disjunction (or) :  $\vee$  (inclusive)   
* Negation (not) :  $\sim$  
* Conditional (if ... then) :  $\longrightarrow$  
* Biconditional (if and only if) :  $\leftrightarrow$  
 $p\rightarrow q$  
* Converse :  $q\rightarrow p$  
* Inverse :  $\sim p\rightarrow\sim q$  
* Contrapositive :  $\sim q\rightarrow p\sim$  

Laws of the algebra of prepositions

* Idempotent Laws  :  $p\vee p\equiv p\ and\ p\wedge p\equiv p$  

* Associative Laws  :  $\left(p\vee q\right)\vee r\equiv p\vee\left(q\vee r\right)\ and\ \left(p\wedge q\right)\wedge r\equiv p\wedge\left(q\wedge r\right)$  
* Commutative Laws  :  $p\vee q\equiv q\vee p\ and\ p\wedge q\equiv q\wedge p$  
* Distributive Laws  :  $\left(p\vee q\right)\wedge\left(p\vee r\right)\equiv p\vee\left(q\wedge r\right)\ and\ p\wedge\left(q\vee r\right)\equiv\left(p\wedge q\right)\vee\left(p\wedge r\right)$  
* Identity Laws  :  $1.\ P\vee F\equiv P\ \ \ 2.\ P\vee T\equiv T\ \ \ 3.\ P\wedge F\equiv F\ \ 4.P\wedge T\equiv P\$  
* Complement Laws  :  $p\vee\sim p\equiv T\ \ \ \ p\wedge\sim p\equiv F\ \ \sim T\equiv F\ \ \sim F\equiv T$  
* Involution Laws  :  $\sim\sim p=p$  
* De Horgan's Laws :  $\sim\left(p\vee q\right)\equiv\sim p\wedge\sim q\ \ \ \ \ \sim\left(p\wedge q\right)\equiv\sim p\vee\sim q$  

FORMAT

Real Number System (Module 1)

 $N=\left\{natural\ numbers\right\}=\left\{1,2,3,4..\right\}$  
 $Z=\left\{intergers\right\}=\left\{...-3,-2,-1,0,1,2,3..\right\}$  
 $W=\left\{whole\ numbers\right\}=\left\{0,1,2,3,...\right\}$  
 $Q=\left\{rational\ nos.\right\}=\left\{\frac{p}{q}\right\}$  where p and q are Z and q  $\ne0$  .
 $Q\ '=\left\{irrational\ nos.\right\}$  

* Closure Property -  $a,b\ \in R\longrightarrow\ a\ +b\in R$  

* Associative Laws -  $a,b,c\in R\rightarrow\ \left(a+b\right)+c=a+\left(b+c\right)\ \&\ \left(a\times b\right)\times c=a\times\left(b\times c\right)$  
* Commutative Laws -  $a,b,c\ \in R\longrightarrow\ a\times\left(b+c\right)=\left(a\times b\right)+\left(a\times c\right)\ \&\ \left(a+b\right)\times c=\left(a\times c\right)+\left(b\times c\right)$  
* Additive Identity - It's 0.  $a\in R\ \rightarrow a\ +0\ =0\ +a=a$  
* Additive Inverse -  $a\in R\ \&\ -a\ \in R\ \longrightarrow\ a+\ \left(-a\right)\ =-a+a\ =0$  
* Multiplicative Inverse -  $a\in R,\ a\ne0\ \rightarrow\ a\times a^{-1}=a^{-1}\times a=1$   0 has no M.I.   Eg.  $2\times\frac{1}{2}=1$  
* Multiplicative Identity - It's 1.  $a\in R\longrightarrow\ 1\times a=a\times1=a$  

Binary Operations

* Commutative - x * y = y * x 
* Associative -  $\left(a\Delta b\right)\Delta c\ =a\Delta\left(b\Delta c\right)$  
* Closed under the operation - x * y  $\in G$   for all x, y  $\in G$  
* Distributive property - a *  $\left(b\Delta c\right)=$  (a * b)  $\Delta$  (a * c) for all  $a,b,c\ \in S$  . * is distributive over  $\Delta$  
* Identity - e * x = x * e = x 

The identity for Addition is 0 and 1 for multiplication.

Surds (Module 1)

*  $\sqrt{xy}=\sqrt{x}\times\sqrt{y}$  
*  $\sqrt{x}\times\sqrt{x}=x$  
*  $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$  
*  $x\sqrt{y}+2x\sqrt{y}=3x\sqrt{y}\left(+\ and\ -\ like\ terms\right)$  
* Rationalizing  $\left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right)=a^2-b$  
* $\left(\sqrt{x}+y\right)^2=\left(\sqrt{x}+y\right)\left(\sqrt{x}+y\right)=x+2y\sqrt{x}+y^2$  

Summation Notation (Module 1)

Standard Results
* If 'c' is a constant.  $\sum_{r=1}^n\ c\ \cdot\ f\left(r\right)=c\cdot\sum_{r=1}^n\ f\left(r\right)$  

*  $\sum_{r=1}^n\left(f\left(r\right)\ \frac{+}{ }\ g\left(r\right)\ \right)\ =\sum_{r=1}^n\ f\left(r\right)\frac{+}{ }\sum_{r=1}^n\ g\left(r\right)$  
*  $\sum_{r=1}^nc=cn$  
*  $\sum_{r=m}^n\ f\left(r\right)=\sum_{r=1}^nf\left(r\right)\ -\sum_{r=1}^{m-1}f\left(r\right)$  
*  $\sum_{r=1}^nr=\frac{1}{2}n\left(n+1\right)\ \ \ \ \ \sum_{r=1}^nr^2=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)\ \ \ \ \ \sum_{r=1}^nr^3=\frac{1}{4}n^2\left(n+1\right)^2$  

Polynomials (Module 1)

General : 
 $a^n-b^n=\left(a-b\right)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+....+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$  
1.  $a^2-b^2=\left(a-b\right)\left(a+b\right)$  
[ $a^2+b^2$  can't be factorized.]

2.  $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\ \ \ \ \ \ \ \ \ \ a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$  

3.   $a^4-b^4=\left(a-b\right)\left(a^3+a^2b+ab^2+b^3\right)=\left(a-b\right)\left[a^2\left(a+b\right)+b^2\left(a+b\right)\right]=\left(a-b\right)\left(a+b\right)\left(a^2+b^2\right)$  $a^4+b^4$  cannot be factorized.

4.  $a^5-b^5=\left(a-b\right)\left(a^4+a^3b+a^2b^2+ab^3+b^4\right)\ \ \ \ \ a^5+b^5=\left(a+b\right)\left(a^4-a^3b+a^2b^2+ab^3+b^4\right)$  

5.  $a^6-b^6=\left(a-b\right)\left(a+b\right)\left(a^2+ab+b^2\right)\left(a^2-ab+b^2\right)$  

 $\left(a+b\right)^3=a^3+3a^2b+3ab^2+b^3$  
 $\left(a+b\right)^2=a^2+2ab+b^2\ \ \ \left(a-b\right)^2=a^2-2ab+b^2$  

Modulus Function (Module 1)

* Definition  $\left|x\right|\ =\left\{\frac{x,\ x\ge0}{x,\ x<0}\right\}$  

Properties :
(a)  $\left|xy\right|\ =\left|x\right|\ \cdot\left|y\right|$  
(b)  $\left|\frac{x}{y}\right|\ =\frac{\left|x\right|}{\left|y\right|},\ y\ne0$  
(c)  $\left|x^n\right|\ =\left|x\right|^n$  
(d)  $-\left|x\right|\ \le x\le\left|x\right|$  
(e)  $\left|x\right|\ =k\Longleftrightarrow x=\frac{+}{ }k$  
(f)  $\left|x\right|\ \le k\Longleftrightarrow-k\le x\le k$  
(g)  $\left|x\right|\ \ge k\Longleftrightarrow x\le-k\ or\ x\ge k$  
(h)  $\left|x\right|\ \le\left|y\right|\Longleftrightarrow x^2\le y^2$  
(i)  $\left|x\right|\ge\left|y\right|\Longleftrightarrow x^2\ge y^2$  
(j)  $\left|x+y\right|\le\left|x\right|+\left|y\right|\ \left(the\ triangle\ inequality\right)$  

Solving inequalities 

(a)  $\left|x\right|<a\ \Longleftrightarrow-a<x<a$  
(b)  $\left|x\right|>a\Longleftrightarrow\ x<-a\ or\ x>a$  
(c)  $\left|x\right|\le a\ \Longleftrightarrow-a\le x\le a$  
(d)  $\left|x\right|\ge a\ \Longleftrightarrow x\le-a\ or\ x\ge a$  
(e)  $\left|x\right|<a\ \Longleftrightarrow x^2<a^2$  
(f)  $\left|x\right|>a\Longleftrightarrow\ x^2>a^2$  
(g)  $\left|x\right|\le a\ \Longleftrightarrow\ x^2\le a^2$  
(h)  $\left|x\right|\ge a\Longleftrightarrow\ x^2\ge a^2$  
 

Cubic Equations (Module 1)

 $ax^3+bx^2+cx+d=0$  
 $\alpha+\beta+\gamma=\frac{-b}{a}$  
 $\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}$  
 $\alpha\beta\gamma=\frac{-d}{a}$  

Short notation :
(a)  $\sum_{ }^{ }\ \alpha=\alpha+\beta+\gamma$        (b)  $\sum_{ }^{ }\alpha\beta=\alpha\beta+\beta\gamma+\gamma\alpha$  
(c)  $\sum_{ }^{ }\ a^2=\alpha^2+\beta^2+\gamma^2$    (d)  $\sum_{ }^{ }\ \alpha^3=\alpha^3+\beta^3+\gamma^3$  
(e)  $\sum_{ }^{ }\alpha\beta\left(\alpha+\beta\right)=\alpha\beta\left(\alpha+\beta\right)+\beta\gamma\left(\beta+\gamma\right)+\alpha\gamma\left(\alpha+\gamma\right)$  

Important results :
(a)  $\sum_{ }^{ }\alpha^2=\left(\sum_{ }^{ }\alpha\right)^2-2\sum_{ }^{ }\alpha\beta$  
(b)  $\sum_{ }^{ }\alpha^3=\left(\sum_{ }^{ }\alpha\right)^3-3\sum_{ }^{ }\alpha\times\sum_{ }^{ }\alpha\beta+\alpha\beta\gamma$  
(c)  $\sum_{ }^{ }\alpha\beta\left(\alpha+\beta\right)=\sum_{ }^{ }\alpha\times\sum_{ }^{ }\alpha\beta-3\alpha\beta\gamma$  
(d)  $\sum_{ }^{ }\alpha^2\beta^2=\left(\sum_{ }^{ }\alpha\beta\right)^2-2\alpha\beta\gamma\left(\sum_{ }^{ }\alpha\right)$  
(e)  $\sum_{ }^{ }\alpha^4=\left(\sum_{ }^{ }a^2\right)^2-2\sum_{ }^{ }a^2\beta^2$  
 

Equation :  $x^3-\left(sum\ of\ new\ roots\right)x^2+\left(sum\ of\ the\ product\ of\ the\ pairs\ of\ roots\right)x-product\ of\ new\ roots\ =0$  

Functions

 $\left[\ \ \ \right]-closed\ interval\ :\ uses\ \le,\ge$  
 $\left(\ \ \right)-open\ interval\ :\ uses\ <,>$  
 $\left[\ \ \right)\ or\ \left(\ \ \right]-Half-open\ :\ \le,>or\ <,\ge\ etc$  
 $\left[x,\frac{+}{ }\infty\right)or\left(\frac{+}{ }\infty,\ x\right]-\ Infinite\ intervals\ :\ where\ x\in Z$  
* Range : the minimum and maximum y values of a function (given its domain)
* Vertical line test tells if a graphed function  is a function
* Horizontal line test determines if a function is injective/one-to-one

Logs

*  $\log_aPQ=\log_aP+\log_aQ$  

*  $\log_a\ \frac{P}{Q}=\log_aP-\log_aQ$  
* $\log_aP^n=n\log_aP$  
*  $\log_a1=0$  
*  $\log_aa=1$  
* Change of Base 
 $\log_ab=\frac{\log_mb}{\log_ma}$