Cartesian Product, Relations, Graphs

8/11/2019 Cartesian Product, Relations, Graphs

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Cartesian Product :

Let A and B be two non empty sets .

The set of all ordered pairs ( a, b) such that

a A and b B is called the cartesian product of

sets A and B. It is denoted by A x B.

e.g. A={ a ,b ,c } B={1,2,3,4}

A x B = { (a,1) , (a,2) , (a,3) ,(a,4) , (b,1) , (b,2) ,

(b,3) ,(b,4) , (c,1) , (c,2) , (c,3) ,(c,4) }


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Relation :

Let A and B be two sets .Then a relation R from A to B is a subset ofA x B.

For e.g. A = {2,4,6} & B = { 3,5,7}

A x B = { (2,3) ,(2,5 ) ,(2,7), (4,3) ,(4,5 ) ,(4,7), (6,3),

(6,5 ) ,(6,7) }

Let R be a relation from A to B defined by aRb : a > b

Then R = { (4,3) ,(6,3) (6,5) }

Thus R is a subset of A x B.


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TYPES OF RELATIONS

Reflexive relation

Symmetric relation

Transitive relation


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REFLEXIVE RELATION

A relation R on a set A is said to be reflexiveif every element of A is related to itself.

SYMMETRIC RELATIONA relation R on a set A is said to be a symmetric relation iff

(a,b) R => (b,a) R for all a, bA

i.e. aRb => bRa for all a, b Y

TRANSITIVE RELATIONLet A be any set. A relation R on A is said to be a

transitive relation iff

(a,b) R and (b,c) R

(a,c) R for all a, b, c A


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Basic points : For a graph of a function y=f(x) , for every x there is one and

only one corresponding value of y.

The graph is set of all the points in the plane of the form ( x ,f(x) ).

The graph of a function therefore has the property that a

vertical line through any point on the x-axis has at most onepoint of intersection with the graph. This is called the verticalline test.

Even Functions : Symmetric about y-axis

Odd Functions : Symmetric in oppositeQuadrants.

Periodicity : The period after which the graph repeats itself ifit is a periodic graph.


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INVERSE TRIGONOMETRIC FUNCTIONS :


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MOD FUNCTION

y = |x|

y = x , for x > 0

y = -x , for x < 0

e.g. when x = 2 , y = 2 but when x = -2 , y = -

(-2) = 2


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Exponential Functions

y = a^x where a >1

e.g. y = 2^x , 3^x , e^x

x = 0 => y = 1x => y

x0 => y0


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LOG Function

y = log x ( where base is a and a>1 or

0 x = 1

y => x

y- => x0


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Polynomial Function :

Graph of a polynomial function is always

continuous .

for e.g. y = x^2 , x^3 etc.


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Greatest Integer Function

Represented as y = [x]

Also called big bracket x. y =[x] => y = -3 -3 x


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Greatest Integer Function


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Signum Function For a graph y= sgn f(x) ,

y = 1 , f(x) >0= 0 , f(x) =0

= -1 , f(x) < 0

e.g. For the graph of y = sgn x , here werefer to the graph of

y = x to plot the graph

of y = sgn x

The graph is also

discontinuous .


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The cost function C gives the cost C(q) ofmanufacturing a quantity q of some good. Alinear cost function has the form

C(q) = mq + b;

where the vertical intercept b is called the

fixed costs, i.e. the costs incurred even ifnothing is produced, and t

he slope m is called the variable costs per unit.


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The function is used to calculate the amount of total consumption in an economy. It is made up ofautonomous consumption that is not influenced by current income and induced consumption thatis influenced by the economy's income level. This function can be where

C = total consumption,

c0 = autonomous consumption (c0 > 0),

c1 is the marginal propensity to consume (ie the induced consumption) (0 < c1 < 1), and

Yd

= disposable income (income after government interventionbenefits, taxes and transferpayments

or Y + (GT)).


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Values of a linear cost function are shown below.

What are the fixed costs and the variable costs

per units? Find a formula for the cost function

Ans

4q +500o

M=4

Q 0 5 10 15

5000 5020 5040 5060 5080


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Example 1

Determine whether the Data in the tablerepresents a linear function.

Step 1

Check the rate of change in the time.

The rate of change is constant, every 10

minutes.

Step 2

Check the rate of change in distance.

This rate of change is NOTconstant.

The ANSWER

Since the rate of change is NOT constantfor both variables, the data does NOTrepresent a linear function.

Use the table to find the

solution

Time

(min)

Distance

biked

(miles)

10 3

20 6

30 10

40 14

50 17

60 19

+10

+10

+10

+10

+10

+3

+4

+4

+3

+2


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Determine whether each equation is a linear

equation

2) 2x2- y = 7Can you write it in standard form?

NO - it has an exponent!

Not linear

3) x = 12x + 0y = 12

A = 1, B = 0, C = 12

Linear!


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FORMULAA :

Finding the slope : The slope mof the line through the

points (x1, y1) and (x2, y2) is given by m = y2-y1/x2-x1,

where x1 is not equal to x2.

Point slope formula :

The equation of a straight line passing through (x1,y1)

with slope a :

y-y1 = a(x-x1)


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Constant Function If a function can be written in the form f(x) = b, where

b is a real number, then we call it a constant function. The graph of a constant function is a horizontal line.

Example 2 : graph of the function f(x) = 5. No matter

what the x value is, the corresponding y value is

always 5.


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PARALLEL AND PERPENDICULAR LINES -

Parallel lines : two lines are parallel if they have the same

slope.

Perpendicular lines : two lines are perpendicular if their

slopes are negative reciprocals of each other.

Horizontal and vertical lines are exceptions to the above.

Recall, that a horizontal line has a slope of zero, wheras a

vertical line has a slope that is undefined.


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Some exceptions :

Variables in the denominator

Variables with exponents

Variables multiplied with other variables.

xy = 12

3y

x

2 3y x

Cartesian Product, Relations, Graphs

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