Algebra New

7/25/2019 Algebra New

1/17

Algebra (Equations,

Expressions and Inequalities)

N.B. equations, expressions and inequalities share much of the same

principles, so there is no need to go in detail on a topic if it has already been

covered in 1 of the main areas.

Algebra de!nition: a set of operations and relations for combining andmanipulating mathematical entities, and the rules that govern them.

First order mathematics is generally concerned with quantities (i.e. numbers) asentities of mathematics; higher order algebra is concerned with other objectssuch as, functions, sets etc.

"he properties of numbers#

$istributive property(pplicable of multiplication) ! this concerns how to dealwith mathematical objects that are placed within parentheses. "he basis of the

distributive property is that multiplication ta#es precedence over addition. $f aparenthesis contains addition between objects, but they are also being multipliedby an e%ternal object, then the multiplication of each term (by the e%ternalobject) ta#es precedence over the addition inside.

&(& ' ) * ' + a(b ' c) ab ' ac

-.. the distributive property is important because it ma#es clear how each termis treated within parentheses, by an e%ternal multiplier ! this is especiallyimportant, as numbers and variables appear to be treated di/erently during

multiplication.

Associative property % this pertains to how parentheses are placed within ane%pression, whereby if you are dealing solely with addition or multiplication, thenno matter where the parentheses are placed the value of the e%pression remainsthe same.

'(& ' ) '( ' &) a '(b ' c) b ' (a ' c) %(0 1 y ) 0(y 1 %)

&ommutative property % this regards the rules when using solely addition orsolely multiplication, whereby changing the ordering of terms does not changethe end value of the e%pression.

& 1 1 &

% ' y y ' %

Additive identity % this refers to the number that when added to any other, itdoes not change the value. "he additive identity is 2.

'ultiplicative identity ! this refers to the number that when multiplied to any

other, it does not change value. "he multiplicative identity is +.


2/17

nverse additive! the inverse additive of a number is its e%act opposite, orspeci3cally the number that when added to it, equals 2. $f

nverse multiplicative! the inverse multiplicative of a number is its reciprocal.

"he property of equality % denotes that e%pressions on either side of an equalsign are equivalent in value.

"he properties of equality#

e*exive property! signi3es that an object is equal to itself.

+ymmetric property! denotes that if a b, then b a. "ransitive property! means that if a b and a c, then b c. +ubstitution property! denotes that if a b, then b can be substituted

for a in any circumstance.

+ome other number properties#

ero product property! denotes that if an object multiplied by another

object is equalled to 2, then either one or both of the objects are 2. For

e%ample % 1 y 2 means either % or y is 2, or it both are 2. "richotomy law ! pertains to the fact that there are & possible relationships

that could possibly occur between two or more objects. "hey are either

equal, more than or less than. 4ssentially, trichotomy denotes that the

relationship between the objects will fall within one of at least one of the &

categories.

Algebra % -irst section

-unctions#

f(%) 5 "his is the general notation of functions but there are others, which will beloo#ed at further on.

function is a mathematical entity that relates input data, to an output.

Functions can be loo#ed at as being constituted of & parts; $nput 6elationship7utput.

"he % within the brac#ets represents the value that is input into the function."he output is placed after the sign, which shows us what happens to the inputafter it goes into the function.

% (input)

f(%) (relationship)

%5! the relationship of this function issquaring; a function ta#es in a value that stems from particular sets, and


3/17

relates it to the output. 8onsequently functions are de3ned as relating aninput to an output.

4%amples of functions (but there are many more):

95

95&'+ ine, 8osine and "angent trigonometric functions.

f is used as the general name for functions, but the name can be anything fromg(%), lol(%) or they can sometimes not have any names at all. For e%ample: %5has no name, but it is a function because it squares the input and gives us asquare number as the output.

f(%) %5&'+ this function suggests that % is the input, the relationship is tocube the input and add +; the output is the %5&'+.

nother way functions can be presented is: y %5 do not ma#e the mista#e

of thin#ing y is the input, because % is the variable being a/ected by theoperation of squaring.

-.. "he di/erence between the input and the output underlines the e/ect thefunction had on the input the function relates the two entities.

Alternative categoriation for di/erent parts of a function#

$omain ! this refers to the set that the input elements (or values) belong.

&odomain ! this refers to an overall set of values that the output elementsbelong, but it should be noted that the codomain consists of elements, which do

not correspond to any of the elements of the input set.ange! the range refers to a subset of the codomain, which consists of theactual elements that the input values correspond with.

"he rules of functions are:

Function must produce an output for every input value of a computable

set; so if it can process + element of a set, then it can process all elements

of that set. For Functions, each and every individual input can only be related to a

single output. 8onsequently functions are considered single-valued.

owever, although no single input element can be related to di/erent outputelements, di/erent input elements can be related to the same output elements.lso, it should be noted that all elements of the input set will be related to someof the elements of the output set, but not all elements of the output set arenecessarily related to all elements of the input set.

For e%ample f(&) *; this function squares the input, which means that &squared can only produce * and no other.

owever it is acceptable if:

f(&) * and f(


4/17

-.. it is important to ma#e note of the rules of functions, because if arelationship does not follow these rules, then it is not a function.

Functions are especially interesting, because how they a/ect an input can bealtered by what #ind of data is input into the function. For e%ample a functionmight square even numbers, yet the same function could set to cube oddnumbers.

-unction Notations#

ome of the di/erent ways that functions can be presented:

Firstly the general way f(%) "ables are another way

s long as the relationshipidenti3ed follows the rules of functions, then it is a function !no matter the presentation.

"he relationship that the table demonstrates is squaring ! whatever value (%) isinput into the function is squared, which gives you the output (y%5).

0ertical line test#

Following that no input value can be related to output values,when plotted ona graph, no two points are allowed to share the same point on the % coordinatei.e. no points are allowed to be vertically aligned ! otherwise the graph doesnot represent a function.

"his is not a function, because the input (%) has more than one possible output.

X: x Y: x2

3 9

1 1

0 0

4 16

-4 16

... ...


5/17

+ets (+et notation)

nother important mathematical principle that is very much entrenched withinfunctions is the idea of sets. Sets are groups of things that are identied (orcategori0ed) together because of certain characteristics they share.

-atural numbers, real numbers and rational numbers are merely some of

the sets that are used in maths.

"he individual objects within the sets are called elements or members.

=hen element of a set are computed into a function, they always produce anoutput that is either from a di/erent set, or the same set.

+ome general information about sets#

ets are unordered, which means the elements in sets do not have to be

outlined in any particular order.

> ?curly brac#ets are generally used to denote and present a set and

its elements >a, b, c?. "here is no set rule when it comes to naming a set, but capital letters are

the common choice when it comes to giving sets a name. @ the Amember ofB symbol denotes that an element belongs to a

particular set. For e%ample, % C >a, b, c, %, y, 0?.

Dmeans Asubset ofB, which is the symbol used to present a set as being

(technically) a strict or proper subset of a particular superset. D E

means that is a subset of E.

this is essentially the opposite of Asubset ofB ! meaning Anot subset ofB

! whereby it denotes that a set is not a subset of another particular set.

Ethis means that is not a subset of E.

Gmeans Asuperset ofB(or strictHproper superset of), whereby a set E

consists of a subset . E G E is a superset of .

Idenotes that a set is not a superset of another, particular set.

"he most commonly used sets are:

Jthe symbol for real numbers.

K the symbol for fractions. L the symbol for the natural (or counting) numbers. M the symbol for integers.

+ome of the notations for ho to present sets#


6/17

et builder notation: >aNa O P?

"he >? denote that the elements inside belong to the set.

"he left side of the vertical line denotes the elements that are

contained within the set. "he vertical line (can be substituted with >a: a O P) stands for Asuch

thatB, which signi3es all elements AaB, belong to the set if they are

equal or greater than P. "he presentation of the setb C =: b Q R?. "his

means b is a whole number that is less than R i.e. >S, P, , &, , and

+?.

-..

+. et


7/17

Eniversal set ! technically spea#ing Auniversal setB refers to an all

Algebra New

Documents

Transcript of Algebra New