Representation of Sets in Mathematics

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Representation of Sets in Mathematics

Topic discussed in this video:

  • Representation of sets in math
  • Roster form / Listing Method
  • Set-builder form / Property Method

 

To understand the representation of sets, let’s get back to its definition.

A set is a well-defined collection of objects of the same kind.

Here, the phrase collection of objects gives us the idea that we can make a list of all the objects. That’s why the first method is known as the listing method. The other name for the listing method is the roster form of the set.

Secondly, the phrase objects of the same kind give us the idea that all the objects in that collection possess a common property and we can represent the set by that common property. That’s why the second method is known as the property method. The other name for the property method is the set-builder form of the set.

Let’s take an example “a collection of the first five natural numbers” to understand the two forms in detail.

As we know, a collection of the first five natural numbers will always have 1, 2, 3, 4, and 5. In roster form, we make a list of these objects like this:
Objects are also known as elements of sets. In roster form, elements are always separated by commas and always enclosed within curly braces {}.
Generally, we usually denote sets by capital letters and elements by small letters. So, if I denote the set by A, then the roster form will look like this:

Before we discuss the set-builder form, there are two important points regarding the roster form that you should remember……

Practice Questions Related to Sets

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Practice Questions related to Sets

Topic discussed in this video:

  • Questions related to sets in Math for better understanding of its definition

 

Question 1. Which of the following are sets? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of the world.
(iii) The collection of all the English alphabet.
(iv) The collection of good students in a class.
(v) The collection of beautiful flowers in a garden.
(vi) A team of eleven best-cricket batsmen of the world.
(vii) The collection of all the natural numbers less than 100.
(viii) A collection of books written by J. K. Rowling.
(ix) The collection of all even integers.
(x) A collection of most dangerous animals of the world.

Base concept of Trigonometry and its Derivation

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Base concept of Trigonometry and its Derivation

Topic discussed in this video:

  • Base Concept of Trigonometry and its derivation

 

One of the beauties of trigonometry is that its rules are equally applicable for all similar triangles because the base concept of trigonometry was derived from the similarity of triangles and in this video, I am going to discuss that base concept with its derivation.

But before I begin our main concept, let’s revise right triangles first.

As we have learned in the right triangles that we have a fixed interior angle of . The side opposite this interior angle is the longest side and it is known as hypotenuse. Because of this fixed interior angle of , the other two interior angles are always complementary. That means,
So, if I assume  as q then  will always be .

As the trigonometry is about the relationships between the interior angles and the side lengths. So, from now onwards, I am going to assume the interior angle q as my main angle for all the concepts and derivations. Also, AB is going to be the opposite side of my main angle and BC is going to be the adjacent side. These special names for all three sides linked with interior angles are going to be very helpful with our main concept and derivation.

Let’s begin the main concept of trigonometry………

Base concept of Trigonometry and its Derivation 1

Vector, Scalar, Physical Quantity, Distance & Displacement

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Vector, Scalar, Physical Quantity, Distance & Displacement

Questions discussed in this video:

  • What is Quantity?
  • What is Physical Quantity?
  • What is Scalar Quantity?
  • What is Vector Quantity?
  • Difference between Scalar and Vector Quantities?
  • Difference between Distance and Displacement?

 

Vector is a type of Physical Quantities, so let’s begin with Physical Quantities.

Quantity word was derived from Greek word “Quantus”, which means “how much” or “how great”. Quantity is simply the amount or the number of something. For example, if you have 10 chocolates then 10 is the quantity of that chocolate. If you have 5 kilograms of Sugar then 5 kilograms is the quantity. So, Quantity means the amount or the number of something.

Now, Physical Quantities are those quantities which can be measured and expressed physically or to be exact, mathematically or in other words, Physical quantities can be expressed by numbers. For example, the love or hatred you have in your heart for someone, can’t be expressed by numbers. You may use gestures to express that love or hatred, but you can’t express that by numbers. So, love and hatred are not Physical Quantities. On the other hand, you can express your weight and your height by numbers, so weight and height are the Physical Quantities.

So, every physical quantity has number and this number is known as the magnitude. This is an important point.

Now, apart from magnitude some physical quantities also have direction. That’s why there are two types of Physical Quantities. First is the Scalar quantity which only has magnitude and second is the Vector Quantity which has magnitude as well as direction. Let’s take an example to understand the both:

In this map, there is location A and location B. If a person wants to travel from location A to location B by a car then he can follow a path like this. This path is known as distance. But, if that person travels with a jetpack then the path will look like this. In this case, he displaced from location A to location B and this path is known as displacement.

Now, in both cases, the path he has covered can be expressed by numbers. So, both cases have magnitudes. Also, both cases have directions but in case of distance, there are multiple directions which can’t be used in our practical life. So, we can say distance only has magnitude. And in case of displacement there is only one direction that can be expressed mathematically and can be used in our practical life. That’s why distance is a scalar quantity with only magnitude and displacement is a vector quantity with magnitude as well as direction.

Scalar quantities and vector quantities both have magnitudes but it’s the direction that makes the difference

Meaning of Set in Mathematics

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Meaning of Set in Mathematics

Questions discussed in this video:

  • What is Set?
  • Use of Set in Practical Life?
  • Types of Sets in practical life?
  • What is set in Mathematics?

 

We often use the word set in our day to day life. For example, we usually have dinner sets in our homes and as the name suggests, a dinner set is a collection of cups, plates, dishes, saucers etc., used for serving food and dinning. So, as with a dinner set, a set is a collection of objects of same kind. And this set can be of two types in our practical lives.

For instance, if I asked you to create a collection of all the songs recorded by Lady Gaga, you can easily do that with the help of some research and the internet. But, instead of a collection of all songs, I asked you to create a collection of top five songs recorded by Lady Gaga. Now, collection of all songs and collection of top five songs they both are collections, they both are sets but there’s a big difference between both.
A collection of all songs will always be the same, whether you will create that collection or I will create that collection, that collection will always have same songs. Yeah, the order in which we collect these songs can be different but the overall collection will always be the same…..

What is Trigonometry and why we chose right triangles to start learning trigonometry?

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What is Trigonometry and why we chose right triangles to start learning trigonometry?

Questions discussed in this video:

  • What is Trigonometry?
  • How to start trigonometry?
  • Which triangle is the best option to start trigonometry?
  • Why we choose right triangles for trigonometry?

 

The word trigonometry was derived from two Greek words “trigonon” and “metron”. Trigonon means Triangle and metron means Measure. So, trigonometry is about measurements in a triangle.
Now, what kind of measurements in a triangle?
As you know, a triangle has three sides with three interior angles and we can measure the side lengths and the interior angles of a triangle. And trigonometry is about these measurements. Or to be precise, trigonometry is about relationships between side lengths and the interior angles of triangle.

Now, the question is which type of triangle is the best option to start learning trigonometry?
As you know, we have different types of triangles in maths. On the basis of lengths of sides, we have equilateral triangles in which all three sides are of equal lengths, then we have isosceles triangles in which two sides are of equal lengths and then we have scalene triangles in which all three sides are of different lengths; all three sides are unequal.
On the basis of interior angles, we have right triangles in which one interior angle is of 90 degrees, then we have acute triangles in which all three angles are smaller then 90 degrees and we have obtuse triangles in which one interior angle is greater than 90 degrees. types of triangles

Now, the classification on the basis of interior angles is quite interesting, because we can easily draw a relationship between right triangles, acute triangles and obtuse triangles, which is not quite easily possible in the case of equilateral triangles, isosceles triangles and scalene triangles………