## 2012-04-23

### The changing face of urban Hyderabad

A few days ago, my family went shopping to the Ameerpet area of Hyderabad. We shopped for about an hour-and-a-half; the time was 17:00. My wife wanted to have some coffee (so did I, in fact). We could not find a place that served coffee in the immediate vicinity. We walked in the general direction of a few restaurants. Thus began an amazing hour of discovery!

We went into the first restaurant that we came across. We seated ourselves at the first table available. Presently, a waiter turned up. Two strong, hot coffees," I said. No, sir," he replied promptly, we don't serve coffee." I was surprised. We picked up the bags, and walked on.

At the next restaurant, we were cautious. We did not go as far as seating ourselves; rather, we waited for a waiter to approach us. Do you serve coffee?" I enquire. We get the same reply, No." I was more surprised. We walked on.

The third restaurant was a familiar one. It has been around for over twenty five years. The last I had visited it, it used to serve coffee, tea and snacks. However, that was several years ago. My four-year-old son complained of hunger by this time. He wanted a pesarattu (a special Telugu dish that is a kind of thin-and-large pancake). I felt that there was a high probability that this restaurant would serve both pesarattu and coffee. So, we climbed up a floor to the restaurant. No, sir. We used to serve South Indian food until about six months ago. We no longer do. Now, we serve Mughalai, Tandoori and Chinese!" I was mildly astonished. My wife and I sighed simultaneously, and we walked on.

My son was very disappointed. As we walked, he was eagerly watching for another restaurant. This time, we had to walk quite some distance before we came across another. Its look made it clear that it was a very non-vegetarian-oriented restaurant. We did not bother to walk in. My wife and I had a quick consultation, and decided to turn around, pass the shopping area, and try in the other direction.

My son's disappointment grew with each passing twenty five metres, or so. He started getting petulant. We negotiated the distance back to the shopping area with some difficulty, coaxing my son along the way. As we walked past that, we soon realised that there were no restaurants within sight! By this time, we had spent close to an hour covering a total of a little over a kilometre, without finding a place that served South Indian snacks and coffee! We resigned, got into the car, and drove back home.

The episode left me wondering, however, about the dramatic transformation that Hyderabad has undergone in the last couple of decades. It is very difficult these days to find decent (or even semi-decent) restaurants that serve Telugu vegetarian food. I have noticed the same trend in Bengaluru too, particularly for supper. A large number of restaurants have colluded to systematically eliminate South Indian menus. A key reason is that Mughalai, Tandoori, Chinese, etc. food is much more expensive. The restaurants earn significantly more per table-hour when they serve them. The constant in-flow of North Indians into Hyderabad has only made it easier for the restaurants to switch over.

Another dimension that has seeped in over the years is that of western fast food (pizzas, burgers, etc.). In the name of maintaining international quality at an international price, the western chains charge ridiculously high prices (by Indian standards) for such fast food. We have to remember, however, that economic liberalisation has placed sudden money and means in the hands of an entire new crop of employees and entrepreneurs (and their pizzas-and-potato-chips brats). India has, consequently, been witnessing rapid changes in urban social patterns. The new-found affluence has resulted in a large number of families dining out several times a week. And, in the name of novelty, a vast majority of them patronise the more expensive varieties. The smaller restaurants, obviously, do not wish to let the opportunity slip by. We see, thus, a steady decline in the number of restaurants serving native food.

Craving for the new often dislodges the old! In this instance, Telugu (South Indian, in general) food and beverages are the casualty!

## 2012-04-07

### Graphs and molecules - 2

Note: This post utilises MathJax to display mathematical notation. It may take a second or two to load, interpret and render; please wait!

## Ordering

The notion of ordering is very intuitive in the context of natural numbers. Indeed, when we learn natural numbers, their representation $$\bbox[1pt]{\{1, 2, 3, 4, \ldots\}}{}$$ itself imprints an ordering relationship in our minds. Soon enough, we learn to assign a sense of relative magnitude to those numbers: 4 is larger than 2, etc. This concept extends naturally to negative numbers and rational numbers too.

### A little rigour

Suppose that we represent the ordering relationship between two elements of a set using the symbol $$\le$$. Then, we can define the properties that a set $$S$$ should satisfy for it to be ordered.

• Reflexivity: $$a \le a\forall a \in S$$
• Antisymmetry: if $$a \le b$$ and $$b \le a$$, then $$a = b\ \forall a, b \in S$$
• Transitivity: if $$a \le b$$ and $$b \le c$$, then $$a \le c\ \forall a, b, c \in S$$

We can readily see that integers and rational numbers satisfy the above properties. Accordingly, we say that integers and rational numbers are ordered, if we assign the meaning smaller than or equal to to the ordering relationship $$\le$$.

In fact, we can see that integers and rational numbers also satisfy an additional property.

• Totality: $$a \le b$$ or $$b \le a\ \forall a, b \in S$$

### A distinction

Totality is a stricter requirement than the preceding three. It mandates that an ordering relationship exist between any and every pair of elements of the set. While reflexivity is easy enough to comprehend, the next two specify the conditions that must hold if the elements concerned do obey an ordering relationship.

It is easy to think of sets that satisfy the former three properties, but without satisfying the last. As an example, let us consider the set $$X = \{1, 2, 3\}$$. Now, let us construct a set of some of its subsets $$S = \{\{2\}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}\}$$. Let us define the ordering relationship $$\le$$ to mean subset of represented by $$\subseteq$$. Exercise: verify that the first three properties hold in $$S$$.

We see that $$\{1, 2\}$$ and $$\{2, 3\}$$ are elements of $$S$$, but neither is a subset of the other.

Therefore, mathematicians distinguish sets satisfying only the first three from those satisfying all the four. The former are said to have partial ordering, and they are sometimes called posets or partially-ordered sets. The latter are said to have total ordering.

## More ordering

Now, let us expand the discussion to include irrational numbers. Do our definitions apply? There is an immediate difficulty: irrational numbers have non-terminating decimal parts! How do we compare two such numbers? How should we define the ordering relationship? The integral part is trivial; it is the decimal part that presents the difficulty.

### Sequence comparison

In order to be able to deal with irrational numbers, we have to introduce an additional notion — sequences. A sequence is a set (finite or infinite) where the relative positions of the elements matter. Another distinction is that elements can repeat, occurring at multiple places. The number of elements in a sequence, if it is finite, is called its length. Thus, sequences can be used to represent the decimal parts of irrational numbers.

Let $$X = \{x_1, x_2, x_3, \ldots\}$$ and $$Y = \{y_1, y_2, y_3, \ldots\}$$ be two sequences. We can define an ordering relationship between sequences as follows. We say $$X \le Y$$ if one of the following holds.

• $$X$$ is finite with a length $$n$$, and $$x_i = y_i\ \forall i \le n$$ and $$y_{n+1}$$ exists.
• $$X$$ and $$Y$$ are infinite, and $$\exists\ n$$ such that $$x_i = y_i\ \forall i \le n$$, and $$x_{n+1} \le y_{n+1}$$.

Armed with the above definition, we can readily see that we can compare two irrational numbers — in fact, any two sequences. Exercise: verify this claim by comparing two irrational numbers and two sequences of non-numerical elements!

### Bottom and top elements

If a set $$S$$ of sequences has an element $$b$$ such that $$b \le s\ \forall s \in S$$, the element $$b$$ is called the bottom element of the set. The element t that we get if we replace $$\le$$ with $$\ge$$ is called the top element of the set. The bottom element and the top are unique in a given set.

In our first example above, $$\{2\}$$ is the bottom element of the set, while $$\{1, 2, 3\}$$ is the top. However, it is important to understand that bottom and top elements may not exist in a given set of sequences. Exercise: think of one such set.

### Minimal and maximal elements

When a set does not have a bottom element, it is yet possible for it to have minimal elements. For an element $$m$$ to be a minimal element of the set $$S$$, $$s \le m \implies s = m$$ should hold. If we replace $$\le$$ with $$\ge$$, we get maximal elements.

Minimal and maximal elements are difficult to establish (and, sometimes, even understand) in the context of infinite sets or complex ordering relationships. The same applies to bottom and top elements, too.

## Conclusion

You may have begun wondering if the title of this post was set by mistake. On the contrary, these concepts are very important to understand before we tackle canonical representation of molecules, ring systems in molecules, etc., which we shall encounter in future posts.