2012-04-07

Graphs and molecules - 2

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If you have not read the previous post in this series, please read it first here.

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.

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