MyMaths MyLife

Some Important definitions we use in sets.

Null Set:
A set is said to be a null set if it has no elements. It is also called an empty set or a void set and is denoted by ø.
Example:
{x | x is a perfect square and 4< x <9}

Finite and infinite sets:
A set ‘A’ is said to be finite if it is either an empty set or contains finite number of elements. Otherwise it is called an infinite set.
Example:
Set of natural numbers less than 100 is finite.

Cardinality of a finite set:
The number of distinct elements in a set is called the cardinality of the set. If a finite set A has n distinct elements, the cardinality of the set is n and is denoted by O(A) or n(A). The cardinality of an empty set is zero.
Example:
Cardinality of A = {a,e,I,o,u} is 5.

A set is a well defined collection objects. The objects of the set are called its elements. Sets are usually denoted by capital letters and the elements of the set are denoted by lower case. If an element x belongs to set A, it is denoted by x Є A. If x is not an element of A, it is denoted by striped(/)Є A.

A set, in general is represented in two forms:
1) In this form, a set is described by actually listing out all the elements. For example, the set of all odd natural numbers less than 10 is represented by {1,3,5,7,9}.

2) In this form, a set is described by a characterizing property. For example, the set of all odd natural numbers less than 10 is represented by {x Є N | x < 10 and x is odd}. The symbol | is read as “such that”.

My opinion is that mathematics is indeed a descriptive language. It is somewhat unique in this role though. The reason for this uniqueness is the fact that when people created this abstract system they wanted to use the most basic elements inherent in our reasoning, rather than just describe the observed. The result of this was a system that had very few and very basic axioms that seemed to match precisely the most basic patterns we observe in the Universe. It also meant that this system could evolve and create many patterns that appear to be greatly similar to those that we find in nature and can describe those.

What do I mean by basic elements ?
Well, one basic element of all reasoning systems we had so far is the existence of separate entities. Others are space, time, laws/relations that control
the entities and possibly more.

Percent implies "for every hundred". This concept is developed to facilitate easier comparison of fractions by equalizing denominator to 100.

Percentages can also be represented as decimal fractions. In such a case it is effectively equivalent to proportion of the original quantity. Any percentage can be expressed as a decimal fraction by dividing the percentage figure by hundred.

Percentages increase or decrease of a quantity is the ratio expressed in percentage of the actual increase or decrease of the quantity and the original amount of the quantity.

1.To convert a common fraction into per cent, we multiply the fraction with 100. And put % symbol to the result.

2.To convert a percent into fraction we must remove % symbol., divide it with 100 and simplify the product.

Factorial is defined for any positive integer. It is denoted by !. Thus “Factorial n” is written as n!. n! is defined as the product of all the integers from 1 to n.

Thus n! = 1.2.3.. ... (n-1),n.

Example 5! = 1*2*3*4*5 = 120

0! is defined to be equal to 1.
Therefore 0! = 1 and 1! = 1

An equation which has the unknown quantity raised only to powers which are whole numbers and the highest power being the square of the unknown quantity, is called a quadratic equation.
The most general form of a quadratic equation is ax^2 + bx + c = 0.
There are two values of x that satisfy such a quadratic equation. These values are called the roots of the quadratic equation.

The roots of the above quadratic equation are given by (-b±√(b^2-4ac))/2a

For ax^2 + bx + c = 0, sum of the roots = -b/a; Product of the roots = c/a

In the previous post we have discussed little bit about algebra. Here take a look about variables and constants.

In arithmetic, the numbers used have definite values. These values do not change. But the letters used in algebra have no particular value and may have any value assigned to them.

Consider,
The perimeter 'p' of a rectangle of length 'l' and breadth 'b' is p = 2(l+b). Here '2' is a fixed number. The letters p, l and b have no fixed value. They can take any positive value depending upon the size of the rectangle.

So, a letter symbol which can take any value of a certain set is called a variable. Above p, l and b are variables. Quantities which have only one fixed value are called constants. Above 2 is a constant.