function and relation

 In the previous post we have discussed about Define one to one correspondence and In today's session we are going to discuss about function and relation.
Function and relation are two different concepts of mathematics. A relation is resembles a function only if it is one – one & onto. Therefore, we can say that all functions are relations but not all relations can be functions. In the Cartesian system we represent the 1^st value of ordered pairs as the x – coordinate and the 2^nd value as the y – coordinate. So, we can define a relation a usual set of orderly pairs and the mapping is not necessarily be one – one & onto.
Suppose we have two sets A = 2, 4, 5, 6, 8, 10 and B = 4, 16, 8, 10, 12. The mapping is done from set A to set B such that the elements in set B consist of multiples in set A. Element 4 of set B has multiples as 2 & 4 in set A. Similarly, for 16 we have 2, 4 & 8, for 8 we have 2, 4 & 8, for 10 we have 2, 5 & 10 and for 12 we have 2, 4 and 6. Thus this mapping is one to many mapping and thus this relation does not represent a function. Thus the ordered pairs we form are: (2, 4), (4, 4), (2, 16), (4, 16), (8, 16), (2, 8), (4, 8), (8, 8), (2, 10), (5, 10), (10, 10), (2, 12), (6, 12) and (4, 14).
To define a functionhttp://en.wikipedia.org/wiki/Function_(mathematics) we must have unique output for every input i.e. one value in range for every value in domain and also all the values of domain and range must be covered. For instance, we have pairs like: (1, 2), (4, 8), (2, 4), (3, 6) and (5, 10).
Next we study about how to find the surface area of a cylinder. For this we have a general formula that is given as follows: 2 pi a² + 2 pi a l. Where, a and l are the radius and height of the cylinder. These concepts are detailed in the  icse syllabus 2013.

Define one to one correspondence

 In the previous post we have discussed about What is Mathematical Induction and In today's session we are going to discuss about Define one to one correspondence.


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One to one correspondence can be defined as property of a set according to which every member of set is related exactly one member of another set. This concept is applied on sets. Set is just a way to store similar kind of values into it and describe the relationship between these values or objects. If we want to describe this concept in the form of simple definition then we can say that it is a situation which occurs only when members of one set are evenly matched with members of another set. According to this one to one correspondence; we need to remember one thing that any member of first set can make a pair with any one member of another set.

Suppose we have two sets which are named as A and B respectively. Set A and B has some members that are shown below:

A = a, b, c, d, e,

B = I, j, k, l, m,

If we follow one to one correspondence concept on above given sets then each and every member of set 'A' can make a pair with one member of Set 'B'.

A → B = a, i, b, j, c, k, d, l, e, m

Above each member of set A makes pair with only one member of Set B. Any member of Set A can make pair with any member of Set B, there is no restrictions to make a pair. One more thing we need to remember is that none of the member should be left as unpaired. It means each and every member must carry a pair value with another set.

In mathematics, concept of Molality Formula can be denoted by numbers of moles of any given substance per liter of solution.

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What is Mathematical Induction

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Mathematical Induction is basically a technique which is used to establish the proof that a given statement is true for all positive integers or natural numbers. We need to prove two statements which are: First statement in the infinite sequence of statements should be true and if anyone statement in infinite sequence of statements is true then next statement will also be true.

Suppose, we have 'n' natural numbers then P(n) holds all natural numbers so:

0 + 1 + 2 + 3 + 4 + 5 +.........+ n = n (n + 1) / 2,

Where P(n) provides a formula to calculate the sum of natural numbers.

Now we use inductive step to prove the above statement. Where P(k) holds natural number then according to statement P (k + 1) also holds then

(0 + 1+ 2+...................+ k) + (k + 1) = [(k+ 1) ((k+ 1) + 1)] / 2,

After using the induction hypothesis:

k (k + 1)/ 2 + (k + 1),

k (k + 1)/ 2 + (k + 1) = [k(k + 1) + 2(k + 1)] / 2,

= (k² + k + 2k + 2) / 2,

= (k + 1) ( k + 2) / 2,

= [(k + 1) ((k + 1) + 1)] / 2.

So the resulting equation shows that P (k+ 1) also holds.

To understand it, we will take an example: We have an equation 23ⁿ – 1 and we have to prove that it is divisible by 11 for all positive integers 'n'.

So, to prove that we put the value of n = 1 then

23¹ – 1 = 22 which is divisible by 11.

Now, according to induction theorem if P (k) holds then P (k + 1) also holds so

23^{k + 1} – 1= 23. 23^k - 1

= 11. 2. 23^k + (23^k – 1),

Here we can see that resulting equation is divisible by 11.

Multiplying and Dividing Fractions is a concept of algebra.

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