Generalizing the product of sets to a universal property

The concept of product and co-product can be generalized to a universal property. However, “how” to do so is given in a rather unclear manner in most books and internet links. An example would be I have somehow patched together a coherent explanation. I hope this helps anyone starting out in Category Theory. LetContinue reading “Generalizing the product of sets to a universal property”

Finally, a valid generalisation of the Third Isomorphism Theorem

Let be a commutative group. Let and be subgroups of . If , then . However, what if ? We will consider the general scenario, where and are any subgroups of , provided is not a proper subgroup of . Then In the case that , we have . I have arrived upon this resultContinue reading “Finally, a valid generalisation of the Third Isomorphism Theorem”

A generalization of the quotient group: an exhaustive analysis of all possible cases

We know that if is a group and is a normal subgroup of , then is a quotient group. Today, we shall try and explore some fundamental questions that have plagued my understanding of Algebra for a long long time. What if is just a subset of , and not necessarily a subgroup? Does meanContinue reading “A generalization of the quotient group: an exhaustive analysis of all possible cases”

Aristotle and I I so remember trying to re-interpret nature based on my own empirical observations and intuition, philosophizing and hatching explanations. I can’t even remember when I started this. Maybe in class 2? When I used to take a sheet of paper and pencil and write and write and write about how clouds form, how insects move,Continue reading “Aristotle and I”

Explaining the beginner’s problems with Topology

When one suddenly starts studying compactness and connectedness and other topological concepts in college, one is likely to get confused. Where did all these concepts come from? Then, seemingly intuitive properties of start being proven using these alien notions. Forming a big picture which includes these concepts seems difficult. One still thinks about the realContinue reading “Explaining the beginner’s problems with Topology”

Choosing a new coordinate system

I often used to wonder why one coordinate system is more appropriate than another in particular situations. For example, when dealing with circular motion, we are often advised to use polar coordinates. I used to wonder why we can’t use only coordinates. My teacher used to say using polar coordinates makes things easier. But IContinue reading “Choosing a new coordinate system”