Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

# Sets In Mathematics

In Mathematics, a set is defined as;
“A collection of well-defined and distinct objects is called set”.
The entries of a set are known as its objects. If you want to get a clear idea about sets, it is necessary that you should have enough idea about well-defined and distinct objects. In Mathematics, well-defined and distinct objects mean those objects which are not equal to other objects. For example, if we say that set of natural numbers, the entries of the natural numbers’ set are 1,2,3,4 and so on. 1, 2, 3 and 4 are well-defined and distinct objects because 1 is not equal to 2 and vice versa. On the other hand, if we say that the set of naughty students in the class, we can’t distinguish between naughty students in the class. In other words, the entries of this set can be equal to each other. As these entries of the set are not well-defined and distinct, therefore, it is not a set.

# How To Represent Sets in Mathematics?

There is also a specific method to represent sets in mathematics. In order to represent sets in mathematics, we have to use ‘Capital alphabets’ i.e A, B, C, D and so on. On the other hand, the entries of a set are written in ‘Small alphabets’ i.e a, b, c, d and so on. Anyhow, the entries of the sets are also written in the form of numerals and symbols etc. These entries of a set are enclosed in the braces or the curly brackets. Some essential examples of the sets are given below;
A = {1, 2, 3, 4}
N = {1, 2, 3, 4, 5…….}
W = {0, 1, 2, 3, 4, 5, 6…….}

# Ways To Represent Sets in Mathematics

There are three possible ways to represent sets in mathematics. These three ways are explained below;

# Tabular Form

The method of representing the set in which we list all the elements of the set and these elements of sets are separated by commas. After listing these elements, we enclose these elements in the curly brackets. Some essential examples of sets that are represented in the tabular form are given below;
A = {6, 7, 8, 9, 10}
E = {2, 4, 6, 8, 10……}
O = {1, 3, 5, 7, 9……}

# Descriptive Form

The process of representing the sets in the form of words is known as descriptive form. Some essential examples of the descriptive form of sets are given below;
N = Set of natural numbers
W = Set of whole numbers
Z = Set of integers
O = Set of odd numbers
E = Set of even numbers

# Set Builder form

The presentation of the sets in the symbolic form is known as set builder form. Some essential examples of the representation of the sets in set builder form are given below;
A = {x | x N ^ x > 10}
B = {x | x W ^ x > 5}
C = {x | x N ^ x > 7}

Related post: Types of triangles with pictures

# Types of Sets in Mathematics

There are different types of sets in mathematics. These types of sets along with their examples are explained below;

# Empty or Null Set

A set which has no element is known as an empty or null set. In order to represent an empty or null set, we use the symbol {} or . We can represent a null or an empty set in the following way;
A =
B = {}

# Finite Set

A set which has the finite number of elements or the limited number of elements is known as a finite set. Some essential examples of finite sets are given below;
A = Set of first ten natural numbers
B = {1, 2, 3, 4, 5}
C = {x | x W ^ x > 5}

# Infinite Set

A set which has the infinite number of elements or the unlimited number of elements is known as an infinite set. Some essential examples of infinite sets are given below;
N = Set of natural numbers
W = {0, 1, 2, 3, 4, 5……..}

# Equivalent Sets

Two sets are called equivalent sets if these sets have the equal number of elements. We can represent two equivalent sets by using the symbol ↔. Some essential examples of equivalent sets are given below;
A = {1, 2, 3}
B = {a, b, c, d}
C = {2, 4, 5, 6}
D = {x, y, z}
A and D are equivalent sets because these sets have an equal number of elements and we can represent these two sets as A ↔ D. On the other hand, B and C are also equivalent sets. Its reason is that these two sets have an equal number of elements and we can represent these two sets as B ↔ C.

# Equal Sets

Two sets are called equal sets if these two sets have equal elements and these elements are also same. In order to represent two equal sets, we use the symbol ‘=’. Some essential examples of equal sets are given below;
A = {1, 2, 3, 4, 5}
B = {a, b, c, d}
C = {2, 3, 4, 5, 1}
D = {a, b, c, d}
A and C are equal sets because these sets have an equal number of elements and these elements are also the same. We can represent these two sets as A = C. On the other hand, B and D are also equal sets because these two sets have an equal number of elements and these elements are also the same. We can also represent these two sets as B = D.

# Singleton Set

A set which has only one element is known as singleton set. Some essential examples of singleton sets are given below;
A = {11}
B = {a}
C = {0}

# Subset

If A and B are two sets and all the elements of set A are present in the set B, set A is called the subset of set B. Example of the subset is given below;
A = {1, 2, 3}
B = {1, 2, 3, 4}
As all the elements of A are present in the B, therefore, A is called the subset of B.

# Proper subset

If A and B are two sets and A is a subset of B and A ≠ B, A is also called a subset of B. An example of the subset is given below;
A = {1, 2}
B = {1, 2, 3, 4}
Now, we can say that A is the subset of B.

# Improper Subset

If A and B are two sets and A is an improper subset of B only either A = B or A contains at least one such element which is not present in set B. Example of an improper subset is given below;
A = {1, 2, 3, 4}
B = {1, 2, 3, 4}
A is an improper subset of B.
A = {1, 2, 3, 4, 7}
B = {1, 2, 3, 4, 5}
A is improper subset of B.

# Power set

A set which contains all the subsets of a set is known as its power set. The power set of A = {1, 2} is given below;
P (A) = { , {1}, {2}, {1,2} }

# Universal Set

A set which is superset of all the sets under consideration is known as a universal set. An example of universal set is given below;
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8}
Now, U is called the universal set.

## Types Of Triangles With Pictures

In our daily life, we have to deal with different kinds of shapes. In these shapes, some shapes are opened and some shapes are closed. In order to differentiate these shapes, we have kept the names of these shapes. Therefore, a closed figure which has three sides that are connected with each other is known as a triangle. Along with three sides, a triangle has also three angles. On the basis of these sides and angles, triangles are classified into different types. Here, we will discuss all types of triangles with the help of pictures.

# Triangles classified by sides

On the basis of sides, triangles are classified into three categories. These three categories of the triangles on the basis of sides are explained below;

# Scalene triangle

A scalene triangle is a such triangle which has different lengths of its sides. It means that there are not congruent sides in the scalene triangle. Along with different lengths of sides, the measure of all the angles of this triangle is also different.

# Isosceles triangle

A triangle which has two congruent sides (same lengths) is known as an isosceles triangle. The two sides of the isosceles triangle which have same lengths are known as legs and the third side of this triangle which has different length is known as base. The angle which is formed by two congruent sides of an isosceles triangle is known as vertex angle and the other two angles are known as base angles. The most important quality of this triangle is that the base angles of this triangle are of equal measurement.

Related post: Difference between rational and irrational numbers

# Equilateral triangle

It is an essential kind of triangle with respect to sides because this triangle has three sides of equal length. In words, we can say that an equilateral triangle is a such a triangle which has all the three sides congruent. We can also say an equilateral triangle as a regular triangle because along with three equal sides, this triangle has also three angles of equal measurement. As the total measurement of all the angles of a triangle is 180°. Therefore, each angle of an equilateral triangle is 60°.

# Triangles classified by angles

On the basis of angles, we can classify these triangles into five different categories. These five categories of the triangles on the basis of angles are explained below;

# Acute triangle

As we know that there are three angles of a triangle. If all the angles of a triangle have less than 90 measurings, this triangle is known as an acute triangle. The most important quality of this triangle is that it can be an isosceles, scalene or equilateral triangle.

# Right-angled triangle

A triangle which has at least one right angle (an angle of exact 90° measurements) is known as a right-angled triangle. A right-angled triangle can be an isosceles or scalene triangle but it can not be an equilateral triangle.

# Obtuse triangle

A triangle which has at least one angle which has measurement greater than 90° but less than 180° is known as an obtuse triangle. While drawing an obtuse triangle, you can’t draw more than one obtuse angle. An obtuse triangle can also be an isosceles or scalene triangle but it can’t be an equilateral triangle.

# Equiangular triangle

A triangle which has all three angles of equal measurement is known as an equiangular triangle. It means that all the angles of a triangle should be 60°. All the equiangular triangles are also equilateral triangles.

# Oblique triangle

A triangle which is not a right-angled triangle is known as an oblique triangle. It means that an oblique triangle can be either an acute triangle, obtuse triangle, equiangular triangle but it can’t be a right-angled triangle.

# Basic properties of triangles

Along with understanding different types of triangles, it is also necessary for us to get an idea about the basic properties of the triangles. These properties are explained below;
1)    According to the angle sum property of a triangle, the sum of all the angles of a triangle should be 180°.
2)    If we add two sides of a triangle, their sum will be greater than the third side of the triangle.
3)    The side which is opposite to the largest angle of the triangle is known as the largest side of the triangle.

## What is The Difference Between Rational and Irrational Numbers

In Mathematics, we have to come across lots of numbers. In these numbers, there come perfect squares, surds, terminating numbers, non-terminating numbers, repeating numbers and non-repeating numbers etc. We usually divide these two numbers into two categories. First category is known as rational numbers and the second category is known as irrational numbers. No doubt, to understand the difference between rational and irrational numbers is a difficult task for the students. Here, we will try to explain the difference between rational and irrational numbers with the help of examples.

## Definition of Rational numbers

In Mathematics, rational numbers are those numbers which are written in the form of p/q such that q≠0. The condition for the rational numbers is that both p and q should belong to Z and Z is a set of integers. The simplest examples of the rational numbers are given below;
a)     1/9
b)    10  or 10/1

## Definition of irrational numbers

The irrational numbers are those numbers which are not written in the form of the p/q. The simplest examples of the irrational numbers are given below;
a)     √3
b)    3/0

## Difference between rational and irrational numbers

Most of the students are not able to understand the difference between the rational and irrational numbers just with the help of their definitions. They require more detail to understand the difference between rational and irrational numbers. The key difference between them is given below;

## Perfect squares are rational numbers and surds are irrational numbers

All the perfect squares are rational numbers and the perfect squares are those numbers which are the squares of an integer. In other words, if we multiply an integer with the same integer, we get a perfect square. The examples of the perfect squares are √ 4, √ 49, √ 324, √ 1089 and √ 1369. After taking the square roots of these perfect squares, we get 2, 7, 18, 33 and 37 respectively. 2, 7, 18, 33 and 37 are all integers.
On the other hand, all the surds are the irrational numbers and the surds are those numbers which are not the squares of an integer. In other words, these are not the multiples of an integer with itself. The examples of the surds are √2, √3 and √7. After taking the square roots of these surds, we get 1.41, 1.73 and 2.64 respectively. 1.41, 1.73 and 2.64 are not integers.

Related post: Types of triangles with pictures

## Terminating decimals are rational numbers

All the terminating decimals are rational numbers. Terminating decimals are those decimals which have finite number of digits after the decimal point. For example, 1.25, 2.34 and 6.94 are all rational numbers. On the other hand, non-terminating decimals are those numbers which have infinite number of digits after the decimal point. For example, 1.235434….., 3.4444… and 6.909090… are all non-terminating decimals. Non-terminating decimals can be rational or irrational. These are explained in the next point.

## Repeating decimals are rational numbers and non-repeating decimals are irrational numbers

All the repeating decimals are the rational numbers and the repeating decimals are those decimals whose digits repeat over and over again without end. The examples of the repeating decimals are .33333333, .222222 and .555555.
On the other hand, all the non-repeating decimals are the irrational numbers and the non-repeating numbers are those digits which don’t repeat over and over again. The examples of the non-repeating decimals are .0435623, .3426452 and .908612.

## Key point

The numbers which are written without denominators are rational numbers. The examples of this kind of numbers are 8 and 9. These numbers are written in the form of p/q as 8/1 and 9/1.
The numbers whose denominators are 0 are called the irrational numbers like 8/0 and 9/0.

## Is ½ or 0.5 rational or irrational number?

0.5 is called the rational number because it can be written in the form of p/q like 5/10. Moreover, it is also a terminating decimal.

## Is Pi (π) rational or irrational number?

Pi (π)  is the irrational number. Its reason is that it gives us non-repeating decimal 3.14159……

## Key point

You can easily express the rational numbers in the fraction form. On the other hand, you can’t express the irrational numbers in the fraction form.

This is the basic difference between the rational and irrational numbers.

## Understand the difference between rational and irrational numbers with the help of practical examples

After understanding the difference between rational and irrational numbers, we try to separate the rational and irrational from given numbers. Separate the rational and irrational numbers from the following numbers;
√5, √25, 5/4, 6/5, √36, √8, 16/3, 6/7
√5 is an irrational number because it is a surd because it is not the square of an integer with itself. √25 is a rational number because it is a square of an integer 5 with itself. 5/4 (1.25) is also a rational number because it is a terminating decimal because it has finite number of digits after the decimal point. 6/5 (1.2) is also a rational decimal because it has also finite number of digits after the decimal point.
√36 is also a rational number because it is a perfect square. √8 is an irrational number because it is a surd. The answer of the fraction 16/3 is 5.33333… It means that it is a repeating decimal. As we know that repeating decimal is also a rational number. The answer of the fraction 6/7 is 0.85714… It means that it is a non-repeating decimal and we have learned that all the non-repeating decimals are irrational numbers.

## Conclusion

At the end, we are able to prepare a chat to clearly understand the difference between rational and irrational numbers.
Rational numbers = Perfect squares + Terminating decimals + Repeating decimals
Irrational numbers = Surds + Non-repeating decimals
You just need to take an overview of a number. If it is a perfect square, terminating decimal or repeating decimal, it means that it a perfect square. On the other hand, if it is a surd or non-repeating decimal, it means that it is an irrational number.