Fruits of procrastination

Month: March, 2014

Inductive sets

Inductive sets are sets such that

1. \emptyset\in A, where A is the inductive set under consideration.
2. x\in A\implies x^+\in A, where x^+=x\cup \{x\}

Natural numbers (here, natural numbers include 0) are those sets that are present in every inductive set. Let us explore this strand of thought more.

Clearly, \emptyset belongs to every inductive set. Hence, it also belongs to the set of natural numbers (or \Bbb{N}). Also, seeing as 0^+,0^{++},\dots belong to every inductive set, all these elements make up the set of natural numbers.

An inductive set that is a proper superset of \Bbb{N} would be \{0,0.5,1,1.5,2,2.5,\dots\}. The successor of every element is there in this set, and it also contains 0. Hence, it is an inductive set.

EDIT: I forgot to mention the motivation behind writing this post. To prove a property for all elements in an inductive set, we have to prove it for all elements without predecessors, and then show that if $a$ satisfies the property, then so does a^+. For example, in the inductive set \{0,0.5,1,1.5,2,2.5,\dots\}, we have to prove the property for 0 and 0.5, and then also prove that a satisfies the property \implies a^+ satisfies the property. Because the only starting point of \Bbb{N} (element without predecessor) is 0, now you  know how and why induction works in natural numbers.

Another rant on undergraduate Algebra

Far too much of Algebra is infested with “equivalence relation” and “well-defined mappings” and what not. Over time, I had grasped their essence, but faltered when trying to fathom their motivation/put them in context. I’m sure you’ve felt the same way in undergraduate Algebra.

Then I made the best decision of my life: I started reading “Elements of Set Theory” by Enderton. I think it is almost criminal if (Indian) colleges let students move on to Algebra and the like without first making them read Set Theory (especially from this book). But more on that later.

1. Equivalence relations- Why do we need equivalence relations? Intuitively speaking, we need them to clump elements together. Elements that we *want* to consider as the same. For example, if I wanted to clump all even numbers together as one element and all odd numbers as another, I would impose the condition “p\sim q iff 2|p+q on \Bbb{N}, the set of natural numbers.

But do we really need equivalence relations to clump elements together which we consider to be the same? No! We can clump elements in whatever fashion we like, and not give a hoot about any specified relations. Heck, we do not even have to ensure that the clumps are disjoint! We can pretty much do anything we want.

However, if we have an equivalence relation, often things become “neat”. Resulting clumps become disjoint, etc. Hence, equivalence relations provide a useful tool, but by no mean the only tools to clump elements together.

2. Well-defined mappings- Often, to check whether a mapping (function, we say in our heads) is well-defined, we check whether the following condition is satisfied: a=b\implies f(a)=f(b). Whut?! This is obvious! Why are we doing all this at all? It was not until recently that I realised the true value of this: this concept is tremendously useful when dealing with equivalence classes (or other kinds of clumps). Say you have a\sim b, where a,b\in P. Also, assume f:P\to Q, where Q is any other set. Then if f satisfies p\sim q\implies f(p)=f(q), we can easily construct a new mapping f^* from f, with f^*:(P/\sim)\to Q.

Essentially, the seemingly useless concept of “well-definedness” starts becoming useful when we have clumps (a more generalised form of equivalence classes).

A rant on the null-set.

This as a post was long due.

What exactly is \emptyset? It is the single most confusing thing I have come across. I am going to try and elaborate its properties. I’m going to build the theory in stages.

1. Nothingness. Let us give it a symbol- \mathfrak{not}. This \mathfrak{not} is an element of EVERY set. \mathfrak{not} is the absence of elements, but it is present in everything. This is a little non-intuitive, if one were to take the physical world as a reference point. For a to be present inside S, there has to be some trance, some sign of the presence of a in S.

Maybe even a space  ” “, showing an absence of elements. But really in set theory, we don’t have spaces of this kind to show an absence of elements. We have no such trace, which is fairly irritating. But one has to get used to it.

2. Now we come to \emptyset. This is NOT equivalent to \mathfrak{not}. \emptyset is not the absence of elements. It is a set containing no element, the operating word being “set”. Technically, \mathfrak{not}\in\emptyset, which makes \emptyset non-empty. However, we work around this by translating to English. When we say \mathfrak{not}\in\emptyset, what we’re really saying is no element belongs to \emptyset, thereby making it an emptyset. See? Clever!

3. Say we have a set A=\{a,b,c\}. Actually, A=\{\mathfrak{not},a,b,c\}. Hence, one element of \mathcal{P}(A) is \{\mathfrak{not}\}=\emptyset, as stated before (by clever linguistics). Hence, \emptyset\in\mathcal{P}(A) for every set A. In fact \mathcal{P}(\emptyset)=\mathcal{P}(\{\mathfrak{not}\})=\{\{\mathfrak{not}\}\}=\{\emptyset\}. Hence, \emptyset\in\mathcal{P}(\emptyset) also.

4. Say we have to construct the set S=\{x\in\Bbb{R}|x=x+1\}. This set contains no element from \mathbb{R}. However, even THIS set has to contain the element \mathfrak{not}. Hence, S=\{\mathfrak{not}\}=\emptyset.

I have sort of developed my own concepts here, and have made set theory more understandable for myself. I hope it helps other readers too.