Relations,Functions and Graphs.

Ordered pairs and Cartesian product

A. Ordered pairs

A pair consists of two elements. Some example of pair are (3,4) (a,b) (d,c). etc. An ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) Ordered pairs are also called 2-tuples, or sequences

Some example of equal and unequal pair

a. (1, 3) and (3, 1) are unequal i.e. (1, 3) ≠ (3, 1).

b. (a, b) and (a, b) are equal i.e. (a, b) = (a, b).

c. (1, a) and (1, x) are unequal i.e. (1, a) ≠ (1, x).

d. (x, x) and (y, y) are unequal i.e. (x, x) ≠ (y, y).

B. Cartesian product

Cartesian product of two non-empty sets A and B is denoted by A×B and defined as the collection of all the ordered pairs (a,b) such that a∈A and b∈B .

A*B = { (a,b):a ∈A,b ∈ B }

Example:

If A = {1, 2, 3} and B = {a, b} find A x A, B x B and B x A

Soln:

Here,

A = {1, 2, 3} and B = {a, b}

So, A x A = {1, 2, 3} x {1, 2, 3}

= {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

Or, B x B = {a, b} x {a, b} = {(a, a), (a, b), (b, a), (b, b)}

And B x A = {a, b) x {1, 2, 3}

= {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

Relation:

A relation is any set of ordered pair (x,y)such that the value of the second coordinate y’ depends on the value of the first coordinate x’ then y is the dependent variable and x is the independent variable.

Function

A function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2.

Composite function

Let f : A → B and g: B → C be the two functions .the function gof : A→ C is called composite function from A to C

Example 3

If  f = { (1,2) ,(2,3) (3,4) } anf g = { (2,a)(4,c) ,(3,b)} , then show that the composite function gof in arrow

diagram and find the it in ordered pair form.

Soln

gof = {(1, a) ,(2,b) , (3,c)}

Types of function

Onto function

A function f from a set X to a set Y is onto, if every element in Y has a corresponding element in X such that f(x) = y.

Into function

A function f from a set X to a set Y is into, if element in Y is proper subset of X

One to one on to function

A function f from a set X to a set Y is One to one on to function, if every element in Y uniquely assign to element of X.

One to one into function

A function f from a set X to a set Y is One to one into function, if element in Y not necessarily assign to element of X.

Many to one on to function

A function f from a set X to a set Y is Many to one on to function, if element in Y has more than one element in x mapped to them.

Many to one into function

A function f from a set X to a set Y is Many to one into function, if element in Y has at least one element which is not mapped to the element of X.

Some simple Algebraic function and their graphs:

1. Linear functions:

These are functions of the form: y = m x + b,

Where m and b are constants

A typical use for linear functions is converting from one quantity or set of units to another. Graphs of these functions are straight lines. m is the slope and b is the y intercept. If m is positive then the line rises to the right and if m is negative then the line falls to the right.

These are functions of the form: y = a x 2 + b x + c,

Where a, b and c are constants

Their graphs are called parabolas. This is the next simplest type of function after the linear function. Falling objects move along parabolic paths. If a is a positive number then the parabola opens upward and if a is a negative number then the parabola opens downward.

3. Power functions:

These are functions of the form:

y = a x b,

Where a and b are constants.

The power b is a positive integer:

When x = 0 these functions are all zero. When x is big and positive they are all big and positive. When x is big and negative then the ones with even powers are big and positive while the ones with odd powers are big and negative.

The power b is a negative integer:

When x = 0 these functions suffer a division by zero and therefore are all infinite. When x is big and positive they are small and positive. When x is big and negative then the ones with even powers are small and positive while the ones with odd powers are small and negative.

The power b is a fraction between 0 and 1:

When x = 0 these functions are all zero. The curves are vertical at the origin and as x increases they increase but curve toward the x axis.

Polynomial functions:

These are functions of the form:

y = an · x n + an −1 · x n −1 + … + a2 · x 2 + a1 · x + a0,

Where an, an −1, … , a2, a1, a0 are constants. Only whole number powers of x are allowed. The highest power of x that occurs is called the degree of the polynomial. The graph shows examples of degree 4 and degree 5 polynomials. The degree gives the maximum number of “ups and downs” that the polynomial can have and also the maximum number of crossings of the x axis that it can have.

Rational functions:

These functions are the ratio of two polynomials. One field of study where they are important is in stability analysis of mechanical and electrical systems (which uses Laplace transforms).

When the polynomial in the denominator is zero then the rational function becomes infinite as indicated by a vertical dotted line (called an asymptote) in its graph. For the example to the right this happens when x = −2 and when x = 7.

When x becomes very large the curve may level off. The curve to the right levels off at y = 5.

The graph to the right shows another example of a rational function. This one has a division by zero at x = 0. It doesn't level off but does approach the straight line y = x when x is large, as indicated by the dotted line (another asymptote).

Exponential functions:

These are functions of the form:

y = a b x,

Where x is in an exponent (not in the base as was the case for power function) and a and b are constants. (Note that only b is raised to the power x; not a.) If the base b is greater than 1 then the result is exponential growth. Many physical quantities grow exponentially (e.g. animal populations and cash in an interest-bearing account).

If the base b is smaller than 1 then the result is exponential decay. Many quantities decay exponentially (e.g. the sunlight reaching a given depth of the ocean and the speed of an object slowing down due to friction).

Logarithmic functions:

There are many equivalent ways to define logarithmic functions. We will define them to be of the form:

y = a ln (x) + b,

Where x is in the natural logarithm and a and b are constants. They are only defined for positive x. For small x they are negative and for large x they are positive but stay small. Logarithmic functions accurately describe the response of the human ear to sounds of varying loudness and the response of the human eye to light of varying brightness.

Sinusoidal functions:

These are functions of the form:

y = a sin (b x + c),

Where a, b and c are constants. Sinusoidal functions are useful for describing anything that has a wave shape with respect to position or time. Examples are waves on the water, the height of the tide during the course of the day and alternating current in electricity. Parameter a (called the amplitude) affects the height of the wave, b (the angular velocity) affects the width of the wave and c (the phase angle) shifts the wave left or right.

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