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Sets, Relations and Functions (Part-2)

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  SETS Lesson 1 How are you, my dear friends? I hope all of you are doing well. With the class 12 pre-board exam coming up- and their joint Entrance examination preparation-most of my day is spent with my students. Preparing mock test papers, checking answer scripts, explaining their mistakes, and so on have kept me quite busy. As a result, my blog post got a little delayed. But as the saying goes, better late than never!  The second part of set theory is Here. In the last post we have discussed-what is set, different types of sets like finite set and infinite sets, null set, singleton set and equal sets. Now we will discuss about Subsets, power set and Universal set. This is all about discussions in lesson 1.  Subsets: If two sets `A` and `B` are such that every element of `A` is also an element of `B`, then `A` is called a subset of `B`.  We write   `A\subseteqB`. If the set `A` is a subset of `B`, then `b` is called supper set of `A` and we write `A\supseteq ...
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Sets, Relations & Functions (Part-1)

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  SETS   Lesson 1 In this lesson, we will discuss sets and different types of sets. First, let us understand what a set is. A set is a well-defined collection of distinct objects or elements. For example: (i) When we refer to living beings (such as human beings, animals, plants) or non-living things (such as numbers, letters, symbols, etc.), we are essentially referring to some collection. (ii) To call a collection “well-defined,” it must be clear whether a particular object belongs to that collection or not. For example: The collection of all natural numbers less than `10` is well-defined, because it is clear that `8` belongs to the collection, whereas `12` does not. The collection of all students of a particular school is well-defined, because it is clear who is a student and who is not. The collection of all capital cities in India of different states is well-defined, because each city is either a capital or not. However, if we consider the collection “the set of all bea...
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Progressions-A.P. and G.P. (Part-6)

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© ® Written by Sankar Ghosh Hello Readers, In my previous blog post (Part 5) , I discussed the sum of n terms of a Geometric Progression (G.P.) in detail. Now, in Part 6 , we will move further and explore three important topics: 1️⃣ Properties of G.P. 2️⃣ Geometric Mean 3️⃣ Infinite Geometric Series As always, the post includes both theoretical explanations and problem-solving and concludes with an exercise section to help you strengthen your understanding. I encourage all readers to attempt the exercises for deeper learning. And of course, feel free to share your doubts in the comment section —I’ll be happy to help you clarify them. Happy Learning! Properties of Geometric Progression Let us discuss some properties of Geometric Progression (1) Square of any term (except first term and last term) is the product of its two adjacent terms. Suppose `t_1,t_2,t_3,...` etc are in G.P. Then `t_2^2=t_1t_3,t_3^2=t_2t_4,t_4^2=t_3t_5,....` etc. (2)  Square of any term (except first te...
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