Relation and Functions

Cartesian products of sets 

Definition : Given two non-empty sets A and B, the set of all ordered pairs (x, y), where x ∈ A and y ∈ B is called Cartesian product of A and B; symbolically, we write A × B = {(x, y) | x ∈ A and y ∈ B} If A = {1, 2, 3} and B = {4, 5}, then A × B = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} and B × A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} (i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal, i.e. (x, y) = (u, v) if and only if x = u, y = v. (ii) If n(A) = p and n (B) = q, then n (A × B) = p × q. (iii) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

Relations 

A Relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product set A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R. For example, the set R = {(1, 2), (– 2, 3), ( 1 2 , 3)} is a relation; the domain of R = {1, – 2, 1 2 } and the range of R = {2, 3}.

 Functions 
A relation f from a set A to a set B is said to be function if every element of set A has one and only one image in set B.

 In other words, a function f is a relation such that no two pairs in the relation has the same first element.

The notation f : X →Y means that f is a function from X to Y. X is called the domain of f and Y is called the co-domain of f. Given an element x ∈ X, there is a unique element y in Y that is related to x. The unique element y to which f relates x is denoted by f (x) and is called f of x, or the value of f at x, or the image of x under f.

The set of all values of f(x) taken together is called the range of f or image of X under f.

Symbolically.

range of f = { y ∈ Y | y = f (x), for some x in X}

Definition : A function which has either R or one of its subsets as its range, is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.



Every function contains two types of variables: independent variables and dependent variables, whose values literally “depend” on the independent variables. For example, in the function y = f(x) = 2x + y, x is independent and y is dependent (in other words, y is a function of x). The valid values for a given independent variable x are collectively called the “domain.” The valid values for a given dependent variable y are collectively called the “range

P-1) Finding the Domain of a Function

1) Determine the type of function you’re working with. 

The domain of the function is all of the x-values (horizontal axis) that will give you a valid y-value output. The function equation may be quadratic, a fraction, or contain roots. To calculate the domain of the function, you must first evaluate the terms within the equation.

• A quadratic function has the form ax² + bx + c:^[2] f(x) = 2x² + 3x + 4
• Examples of functions with fractions include: f(x) = (¹/_x), f(x) = ^{(x + 1)}/_{(x - 1)}, etc.
• Functions with a root include: f(x) = √x, f(x) = √(x² + 1), f(x) = √-x, etc.

2) Write the domain with proper notation. Writing the domain of a function involves the use of both brackets [,] and parentheses (,). You use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. The letter U indicates a union that connects parts of a domain that may be separated by a gap.^[3]

• For example, a domain of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10.
• Always use parentheses if you are a using the infinity symbol, ∞. This is because infinity is a concept and not a number.

3) Draw a graph of the quadratic equation. Quadratic equations make a parabolic graph that either points up or down. Given that the parabola will continue infinitely outward on the x-axis, the domain of most quadratic function is all real numbers. Stated another way, a quadratic equation encompasses all of the x-values on the number line, making its domain R (the symbol for all real numbers).^[4]

• To get an idea of the function choose any x-value and plug it into the function. Solving the function with this x-value will output a y-value. These x- and y-values are a coordinate (x, y) of the graph of the function.
• Plot this coordinate and repeat the process with another x-value.
• Plotting a few values in this fashion should give you a general idea of shape of the quadratic function.

4) Set the denominator equal to zero, if it’s a fraction. When working with a fraction, you can never divide by zero. By setting the denominator equal to zero and solving for x, you can calculate the values that will be excluded in the function.^[5]

• For example: Identify the domain of the function f(x) = ^{(x + 1)}/_{(x - 1)}.
• The denominator of this function is (x - 1).
• Set it equal to zero and solve for x: x – 1 = 0, x = 1.
• Write the domain: The domain of this function cannot include 1, but includes all real numbers except 1; therefore, the domain is (-∞, 1) U (1, ∞).
• (-∞, 1) U (1, ∞) can be read as the set of all real numbers excluding 1.The infinity symbol, ∞, represents all real numbers. In this case, all real numbers greater than 1 and less than one are included in the domain.

5) Set the terms inside the radical to be greater than or equal to zero, if there’s a root function. You cannot take the square root of a negative number; therefore, any x-value that leads to a negative number must be excluded from the domain of that function.^[6]

• For example: Identify the domain of the function f(x) = √(x + 3).
• The terms within the radical are (x + 3).
• Set them greater than or equal to zero: (x + 3) ≥ 0.
• Solve for x: x ≥ -3.
• The domain of this function includes all real numbers greater than or equal to -3; therefore, the domain is [-3, ∞).

P-2) Finding the Range of a Quadratic Function

1) Confirm that you have a quadratic function. A quadratic function has the form ax² + bx + c: f(x) = 2x² + 3x + 4. The shape of a quadratic function on a graph is parabola pointing up or down. There are different methods to calculating the range of a function depending on the type you are working with.

the easiest way to identify the range of other functions, such as root and fraction functions, is to draw the graph of the function using a graphing calculator.

2) Find the x-value of the vertex of the function. The vertex of a quadratic function is the tip of the parabola. Remember, a quadratic equation is of the form ax² + bx + c. To find the x-coordinate use the equation x = -b/2a. This equation is a derivative of the basic quadratic function which represents the equation with a zero slope (at the vertex of the graph, the slope of the function is zero).

• For example, find the range of 3x² + 6x -2.
• Calculate x-coordinate of vertex: x = -b/2a = -6/(2*3) = -1

3) Calculate the y-value of the vertex of the function. Plug the x-coordinate into the function to calculate the corresponding y-value of the vertex. This y-value denotes the edge of your range for the function.

• Calculate y-coordinate: y = 3x² + 6x – 2 = 3(-1)² + 6(-1) -2 = -5.
• The vertex of this function is (-1, -5).

4) Determine the direction of the parabola by plugging in at least one more x-value. Choose any other x-value and plug it into the function to calculate the corresponding y-value. If the y-value is above the vertex, the parabola continues to +∞. If the y-value is below the vertex, the parabola continues to -∞.

• Use the x-value -2: y = 3x² + 6x – 2 = y = 3(-2)² + 6(-2) – 2 = 12 -12 -2 = -2.
• This yields the coordinate (-2, -2).
• This coordinate tells you that the parabola continues above the vertex (-1, -5); therefore, the range encompasses all y-values above -5.
• The range of this function is [-5, ∞)

5) Write the range with proper notation. Like the domain, the range is written with the same notation. Use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. The letter U indicates a union that connects parts of a domain that may be separated by a gap.^[9]

• For example, a range of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10.
• Always use parentheses if you are a using the infinity symbol, ∞

P-3) Finding the Range of a Function Graphically

1) Graph the function. Oftentimes, it is easiest to determine the range of a function by simply graphing it. Many root functions have a range of (-∞, 0] or [0, +∞) because the vertex of the sideways parabola is on the horizontal, x-axis. In this case, the function encompasses all of the positive y-values if the parabola goes up, or all of the negative y-values if the parabola goes down. Fraction functions will have asymptotes that define the range.^[10]

• Some root functions will start above or below the x-axis. In this case, the range is determined by the point the root function starts. If the parabola starts at y = -4 and goes up, then the range is [-4, +∞).
• The easiest way to graph a function is to use a graphing program or a graphing calculator.
• If you do not have a graphing calculator, you can draw a rough sketch of a graph by plugging x-values into the function and getting the corresponding y-values. Plot these coordinates on the graph to get an idea of the shape of the graph.

2) Find the minimum of the function. Once you have graphed the function, you should be able to clearly see the lowest point of the graph. If there is no obvious minimum, know that some functions will continue on to -∞.

• A fraction function will include all points except those at the asymptote. They often have ranges such as (-∞, 6) U (6, ∞).

3) Determine the maximum of the function. Again, after graphing, you should be able to identify the maximum point of the function. Some functions will continue on to +∞ and therefore, will not have a maximum.

4) Write the range with proper notation. Like the domain, the range is written with the same notation. Use a bracket when the number is included in the domain and use a parenthesis when the domain does not include the number. The letter U indicates a union that connects parts of a domain that may be separated by a gap.^[11]

• For example, a range of [-2, 10) U (10, 2] includes -2 and 2, but does not include number 10.
• Always use parentheses if you are a using the infinity symbol, ∞.

Questions on License plate

1). A license plate consists of a combination of 6 digits or letters. All numbers (0-9) and all 26 letters may be used. How many unique license plates are there?

2). A license plate has 3 letters and 3 digits in that order. A witness to a hit and run accident saw the first 2 letters and the last digit. If the letters and digits can be repeated, how many license plates must be checked by the police to find the culprit?

3). A small nation issues license plates that consist of just one number (selected from the digits 0 through 9, inclusive) and four letters, selected from a 20-letter alphabet. Repeats are permitted. However, there is one four-letter combination that is not allowed to appear on license plates. How many allowable license plate combinations exist?

4). In State X, all vehicle license plates have 2 letters from the 26 letters of the alphabet followed by 3 one digit numbers. How many different license plates can State X have if repetition of letters and numbers is allowed?

5). A license plate has 9 digits. You cannot repeat a digit.How many combination?

6). How many automobile license plates can be made, if each plate contains two different letters followed by three different digits ?

Chess related questions

Q.1) Find the number of squares which can be formed from 8 cm x 8 cm chessboard?
Q.2) Find the number of rectangles which can be formed from 8 cm x 8 cm chessboard?
Q.3) Find the number of pure rectangles which can be formed from 8 cm x 8 cm chessboard?
Q.4) Find the number of parallelograms which can be formed from 8 cm x 8 cm chessboard?
Q.5) Two squares are chosen at random from small squares drawn on a chess board. Find the number of ways in which 2 squares can be chosen such that they have exactly one corner in common?
Q.6) Find the number of ways in which a white and a black square on a chess board be chosen so that two squares do not belong to same row or column?
Q.7) In how many ways can you place 2 rooks on 8*8 chessboard such that they are not in attacking positions?
Q.8)  If two squares are chosen on a 8*8 chessboard, what is the probability that they have one side in common?
Q.9) If two squares are chosen on a 8*8 chessboard, what is the probability that they have one vertex in common?

Concepts of Number System

The number system mainly into classified into 8 types.

1. Complex numbers
2. Imaginary numbers
3. Real numbers
4. Rational numbers
5. Irrational numbers
6. Integers
7. Whole numbers
8. Natural numbers

1. Complex numbers :

Every number in number system taken as a complex number

2. Imaginary Number :

A number does not exist in the number line is called imaginary number. For example square root of negative numbers are imaginary numbers.

It is denoted by ” i ”.

i.e √-1 = i

i2 = – 1

So there is no real number i that satisfies the above equation.  The quantity ” i” is called the unit imaginary number.

3. Real numbers :

All numbers that can be represented on the number line are called real numbers.

The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line.

Real Numbers are denoted by “R”.

4. Rational numbers :

A rational number is defined as number of the form x/y where x and y are integers and Y # 0.

i.e Any number which can be expressed as in the form of p/q where “p” and “q” are the integers and q # 0

The set of rational numbers encloses the set of integers and fractions.

The rational numbers that are not integral will have decimal values. These values can be of two types

Terminating decimal fractions (finite decimal factors): For example 1/5 = 0.5 , 13/5 = 2.6.

Non Terminating decimal fractions : The non terminating decimal fractions having two types.

• Non terminating periodic fractions
• Non terminating non periodic fractions

Non terminating periodic fractions : These are non terminating decimal fractions of the type   a. b1b2b3b4b5  …..bmb1b2b3b4b5  …..bm.

19/6 = 3.16666666…..

18/7 = 2.57142857142857…….

21/9= 2.3333…….

Non terminating non periodic fractions : These are non terminating and there is no periodic decimal places for that number . i.e a. b1b2b3b4b5  …..bmc1c2………

for example 6.789542587436512……….

So from above terminating and non terminating periodic fraction numbers belongs to rational numbers.

5. Irrational numbers :

An Irrational numbers are non terminating and non periodic fractions. i.e irrational number is a number that cannot be written as a ratio x/y form (or fraction).  In decimal form, is never ends or repeats.

Examples for irrational numbers are √2 = 1.414213……, π = 3.14159265…….,  √3, √5 etc

6. Integers ( numbers having no decimals )

All numbers that do not having the decimal places in them are called integers.

All whole numbers including Negative number + Positive number

Z = {∞…..-5,-4,-3,-2,-1,0,1,2,3,4,5….∞}

i.e the integer it may positive or negative or zero.

The set of integers generally written  Z for short.

Any integers are added, subtracted, or multiplied the result is always is an integer.

When any integers multiplied , each of the multiplied integer is called a factor or divisor of the resulting product.

 7. Whole numbers :

The set of whole numbers means narrator numbers and “0”

Whole numbers = W = { 0,1,2,3,4,5,6,7,8,…………..∞}

8. Natural Numbers:

The counting numbers start with 1 and their end is not defined. Generally it is  denoted by “N”

i.e N ={1,2,3,4 ………………………….∞}

Another Some important number system :

• Negative integers
• Non Negative integers
• Positive integers
• Non positive integers
• Prime numbers
• Composite numbers
• Co prime numbers
• Even and Odd numbers
• Perfect number.

Negative integers:

The integer is less than zero than it is called negative integer.

i.e the set of integers { -1,-2,-3,-4,-5………….∞} is called negative integers.

The product of any two negative integers is always positive integer.

Non Negative integers:

The set of numbers Zero and natural numbers  is called non negative integers.

i.e {01,2,3,4 ………………………….∞}

Positive integers:

The integer is greater than zero than it is called positive integer. The set of natural numbers  is called non negative integers.

The product of any two positive integers is always positive integer.

The product of any two positive and negative integers is always negative integer.

i.e the set of numbers {01,2,3,4 ………………………….∞}

Non positive integers:

The integer is less than zero and include zero than it is called non positive integer.

The set of numbers { 0,-1,-2,-3,-4,-5………….∞} is called non positive integers

Prime numbers:

A natural number larger than unity is a prime number and it does not having any other divisors except for unity and itself is called a prime number

For example : 2,3,5,7,11,13,17,19…………..

Crucial facts:-

• “1 ” is not a prime number.
• “2 ” is the only even and lowest prime number.
• Any prime number more than 6 than, it comes reminder 1 or 5 when that number is division by 6.
• Any prime number more than 5 , that prime number square  division by 24  than reminder comes always 1
• If p and q are any two prime numbers than p2+q2 and p2 -q2are composite numbers.
• Prime number exactly having two different factors.
• No. of prime numbers from 1 to 50 = 15
• No. of prime numbers from 1 to 100 =25.
• No. of prime numbers from 1 to 200 =46
• No. of prime numbers from 1 to 1000 =168

Composite numbers:

A number having one more divisor apart from 1 and itself is called composite number.

i.e The composite number having atleast three divisors.

In the mathematical term any composite number ” n ” can be expressed as

n = m1a x m2b x m3c x m4d

Here m1,  m2, m3 , m4, are the prime numbers and a.b,c,d are the natural numbers.

For example 80 = 51 x 24

Co prime numbers:

Co prime numbers means its having the atleast two numbers and its HCF are  “1 ‘

i.e When two or more numbers having no common prime factors apart from the number ” 1″ , they are called co-prime  or relatively prime to each other.

Crucial facts:-

• Two consecutive odd numbers  are always co-prime numbers  ( Ex : 9 & 11 ,  15&17, 21&23 … etc )
• Two prime numbers are always prime numbers  ( Ex : 3& 7, 11&17, …..etc )
• One prime number and another composite number (Such composite number is not a multiple of the prime number ) are always co-prime numbers. This rule exception only for 17 &51. ( Ex : 3 & 16, 4& 9, ….. etc )
• Three or more numbers being co-prime with each other means that all possible pairs of the numbers would be co- prime with each other.  ( Ex: Thus 51, 53 and 55 are co prime each other than the pairs 51&53, 53&55 ,  51&55 are also co-prime numbers)
• Three consecutive odd numbers are always being co-prime numbers. ( Ex : 21,23& 25,  31,33&35 ……… etc.)

Even and Odd numbers:

A number which is a multiple of the number “2” than it is called even number. The even number can be represented by “2n” . here n is natural number.

A number which is not multiple of the number “2” than it is called odd number. The odd number can be represented by “2n+1” . here n is natural number.

Operation of Even and Odd numbers:

Odd x Odd = Odd

Odd + Odd = Even

Odd – Odd = Even

Odd /  Odd = Odd

Even + Even = Even

Even x Even = Even

Even – even = Even

Even / Even = Even or Odd

Odd x Even = Even

Odd + Even = Odd

Even/ Odd = Even

Odd / Even = Not divisible

Perfect number:

Any number  is called perfect number, if the sum of all its factors excepting itself is equal to the number itself .

For example take a numbers 6. Factors of 6 are 1,2,3,6

Sum of the factors except itself = 1+2+3 = 6

So the number ” 6 ” is called perfect number.

Sum of natural, odd & even numbers

1)  Sum of “n” natural numbers = n (n+1)/2

2)  Sum of “n” natural even numbers = n (n+1)

3)  Sum of “n” natural odd numbers = n²

= 

EduCAT: Numbers : Divisibility Osculation

EduCAT: Numbers : Divisibility Osculation: finding the divisibility of a number by 7,13,17,19.the method is known as osculator method.first we have to know the concept of osculator...

Numbers : Divisibility Osculation

finding the divisibility of a number by 7,13,17,19.the method is known as osculator method.first we have to know the concept of osculator.the osculator of 7 is 2 ( 7* 3 = 21 = 20+1 here we are adding 1 so we need to consider the osculator as negative osculator )

 similarly for 13 ,13 * 3 = 39 = 40-1 so the osculator is 40 and is one more osculator and the value is 4 and for 17 ,17 * 3 = 51 = 50+1 so again the osculator is negative and it is 5 and for 19 it is one more osculator  19 = 20-1 ,so the osculator is 2 .

if you didn't understand the concept of osculator just remember these

for 7 osculator = 2 , sign = ' - '

for 13 osculator = 4 , sign = ' + '

for 17 osculator = 5 , sign = ' - '

for 19 osculator = 2 , sign = ' + '

lets proceed with an example

55277838 is considered and to check it's divisibility by 7.

5527783 8  : 5527783 - 8 *2 =5527767 ( here we have taken the units digit and multiplied it by osculator 2 and subtracted it from the remaining part as the sign is negative as suggested above we have subtracted )

the next step is the following is repeated for the resulting value in the above step

552776 7   : 552776 - 7 *2 = 552762

55276 2 : 55276 - 2 * 2 = 55272

5527 2 : 5527 - 2 *2 = 5523

552 3 : 552 - 3 *2 = 546

54 6 : 54 - 6*2 = 42 as 42 is divisible by 7 so the number is divisible by