In order to solve the following tutorial sheet, it helps to solve/understand and then apply the following beautiful cyclic relations or identities:
(Note if these look new to you, then you need to check the truth of all them; if all are v v familiar to you, just go ahead and crack the tutorial sheet below):
Core Identities in Cyclic Expressions:
Solve or simplify the following:
I) Prove that every function can be represented as a sum of an even function and an odd function.
II)Let A, B, C be subsets of a set S. Prove the following statements and illustrate them with Venn Diagrams:
2a) The famous DeMorgan’s laws in their basic forms: and . Assume that both sets A and B are subsets of Set S. In words, the first is: union of complements is the complement of intersection; the second is: intersection of two complements is the complement of the union of the two sets.
Let us say that we need to prove: .
Proof: It must be shown that the two sets have the same elements; in other words, that each element of the set on LHS is an element of the set on RHS and vice-versa.
If , then and . This means that , and and . Since and , hence . Hence, .
Conversely, if , then and . Therefore, and . Thus, and , so that . QED.
III) Prove that if I and S are sets and if for each , we have , then .
It must be shown that each element of the set on the LHS is an element of the set on RHS, and vice-versa.
If , then and . Therefore, , for at least one . Thus, , so that .
Conversely, if , then for some , we have . Thus, and . Since , we have . Therefore, . QED.
IV) If A, B and C are sets, show that :
V) Let I be a nonempty set and for each let be a set. Prove that
5a) for any set B, we have :
5b) if each is a subset of a given set S, then
VI) Prove that if , , and are functions, then :
VII) Let be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:
7i) ; in words, image of union of two sets is the union of two images;
7ii) ; in words, image of intersection of two sets is a subset of the intersection of the two images;
7iii) ; in words, the inverse image of the union of two sets is the union of the images of the two sets.
7iv) ; in words, the inverse image of intersection of two sets is intersection of the two inverse images.
7v) ; in words, the inverse of the image of a set contains the set itself.
7vi) ; in words, the image of an inverse image of a set is a subset of that set.
For questions 8 and 9, we can assume that the function f is and a set A lies in domain X and a set C lies in co-domain Y.
8) Prove that a function f is 1-1 if and only if for all ; in words, a function sends different inputs to different outputs iff a set in its domain is the same as the inverse of the image of that set itself.
9) Prove that a function f is onto if and only if for all ; in words, the image of a domain is equal to whole co-domain (which is same as range) iff a set in its domain is the same as the image of the inverse image of that set.
We shall now complicate the situation a little bit. Let us ask for the combinations of the symbols with repetitions of each symbol allowed once more in the combinations. For example, let there be only two symbols . Let us look for combinations of the form:
, , , , , , ,
where, in each combination, each symbol may occur once, twice, or not at all. The OGF for this can be constructed by reasoning as follows: the choices for are not-, once, twice. This is represented by the factor . Similarly, the possible choices for correspond to the factor . So, the required OGF is
On expansion, this gives :
Note that if we omit the term 1 (which corresponds to not choosing any ), the other 8 terms correspond to the 8 different combinations listed in (*). Also, observe that the exponent r of the tells us that the coefficient of has the list or inventory of the r-combinations (under the required specification — in this case, with the restriction on repetitions of symbols) in it:
In the light of the foregoing discussion, let us now take up the following question again: in how many ways, can a total of 16 be obtained by rolling 4 dice once?; the contribution of each die to the total is either a “1” or a “2” or a “3” or a “4” or a “5” or a “6”. The contributions from each of the 4 dice have to be added to get the total — in this case, 16. So, if we write:
as the factor corresponding to the first die, the factors corresponding to the other three dice are exactly the same. The product of these factors would be:
Each term in the expansion of this would be a power of t, and the exponent k of such a term is nothing but the total of the four contributions which went into it. The number of times a term can be obtained is exactly the number of times k can be obtained as a total on a throw of the four dice. So, if is the coefficient of in the expansion, is the answer for the above question. Further, since (*) simplifies to , it follows that the answer for the above question tallies with the coefficient specified in the following next question: calculate the coefficient of in .6
Now, consider the following problem: Express the number of ways of obtaining a total of n by rolling p dice, as a certain coefficient in a suitable product of binomial expansions in powers of t. [ this in turn, is related to the observation that the number of ways a total of 16 can be obtained by rolling 4 dice once is the same as the coefficient of in ]:
So, we get that coefficient of in
Let us take an example from a graphical enumeration:
A is a set V of vertices a, b, c, …, together with a set of If is considered the same as , we say the graph is . Otherwise, the graph is said to be , and we say ‘ has a direction from a to b’. The edge is called a loop. The graph is said to be of order .
If the edge-set E is allowed to be a multiset, that is, if an edge is allowed to occur more than once, (and, this may be called a ‘multiple edge’), we refer to the graph as a general graph.
If and denote the numbers of undirected (respectively, directed) loopless graphs of order 5, with n edges, none of them a multiple edge, find the series and .
Applying our recently developed techniques to the above question, a graph of 5 specified vertices is uniquely determined once you specify which pairs of vertices are ‘joined’. Suppose we are required to consider only graphs with 4 edges. This would need four pairs of vertices to be selected out of the total of equal to 10 pairs that are available. So selection of pairs of vertices could be made in ways. Each such selection corresponds to one unique graph, with the selected pairs being considered as edges. More informally, having selected a certain pairs of vertices, imagine that the vertices are represented by dots in a diagram and join the vertices of each selected pair by a running line. Then, the “graph” becomes a “visible” object. Note that the number of graphs is just the number of selections of pairs of vertices. Hence, .
Or, one could approach this problem in a different way. Imagine that you have a complete graph on 5 vertices — the “completeness” here means that every possible pair of vertices has been joined by an edge. From the complete graph which has 10 edges, one has to choose 4 edges — any four, for that matter — in order to get a graph as required by the problem.
On the same lines for a directed graph, one has a universe of 10 by 2, that is, 29 edges to choose from, for, each pair x,y gives rise to two possible edges and . Hence,
Thus, the counting series for labelled graphs on 5 vertices is
and the counting series for directed labelled graphs on 5 vertices is
Finally, the OGF for increasing words on an alphabet with is
The corresponding OE is which is nothing but (this explains the following problem: Verify that the number of increasing words of length 10 out of the alphabet with is the coefficient of in ).
We will continue this detailed discussion/exploration in the next article.
Until then aufwiedersehen,
Thanks Dr. Gowers’. These are invaluable insights into basics. Thanks for giving so much of your time.
Thanks a lot Prof. Gowers! Math should be sans ambiguities as far as possible…!
I hope my students and readers can appreciate the details in this blog article of Prof. Gowers.
GENERATING FUNCTIONS and RECURRENCE RELATIONS:
The concept of a generating function is one of the most useful and basic concepts in the theory of combinatorial enumeration. If we want to count a collection of objects that depend in some way on n objects and if the desired value is say, , then a series in powers of t such as is called a generating function for . The generating functions arise in two different ways. One is from the investigation of recurrence relations and another is more straightforward: the generating functions arise as counting devices, different terms being specifically included to account for specific situations which we wished to count or ignore. This is a very fundamental, though difficult, technique in combinatorics. It requires considerable ingenuity for its success. We will have a look at the bare basics of such stuff.
We start here with the common knowledge:
….(2i) where sum of the products of the ‘s taken r at a time. …(2ii)
Incidentally, the ‘s thus defined in (2ii) are called the elementary symmetric functions associated with the ‘s. We will re-visit these functions later.
Let us consider the algebraic identity (2i) from a combinatorial viewpoint. The explicit expansion in powers of t of the RHS of (2i) is symbolically a listing of the various combinations of the ‘s in the following sense:
represents all the 1-combinations of the ‘s
represents all the 2-combinations of the ‘s
and so on.
In other words, if we want the r-combinations of the ‘s, we have to look only at the coefficients of . Since the LHS of (2i) is an expression which is easily constructed and its expansion generates the combinations in the said manner,we say that the LHS of (2i) is a Generating Function (GF) for the combinations of the ‘s. It may happen that we are interested only in the number of combinations and not in a listing or inventory of them. Then, we need to look for only the number of terms in each coefficient above and this number will be easily obtained if we set each as 1. Thus, the GF for the number of combinations is n times;
and this is nothing but . We already know that the expansion of this gives as the coefficient of and this tallies with the fact that the number of r-combinations of the ‘s is . Abstracting these ideas, we make the following definition:
The Ordinary Generating Function (OGF) for a sequence of symbolic expressions is the series
If is a number which counts a certain type of combinations or permutations, the series is called the Ordinary Enumeration (OE) or counting series for for
The OGF for the combinations of five symbols a, b, c, d, e is
The OE for the same is . The coefficient of in the first expression is
(*) abcd+abce+ abde+acde+bcde.
The coefficient of in the second expression is , that is, 5 and this is the number of terms in (*).
The OGF for the elementary symmetric functions in the symbols is ….(2iv)
This is exactly the algebraic result with which we started this section.
The fact that the series on the HRS of (2iii) is an infinite series should not bother us with questions of convergence and the like. For, throughout (combinatorics) we shall be working only in the framework of “formal power series” which we now elaborate.
*THE ALGEBRA OF FORMAL POWER SERIES*
The vector space of infinite sequences of real numbers is well-known. If and are two sequences, their sum is the sequence , and a scalar multiple of the sequence is . We now identify the sequence with with the “formal” series
where only means the following:
In the same way, , where corresponds to the formal series:
we define: , and .
The set of all power series f now becomes a vector space isomorphic to the space of infinite sequences of real numbers. The zero element of this space is the series with every coefficient zero.
Now, let us define a product of two formal power series. Given f and g as above, we write where
, where .
The multiplication is associative, commutative, and also distributive wrt addition. (the students/readers can take up this as an appetizer exercise !!) In fact, the set of all formal power series becomes an algebra. It is called the algebra of formal power series over the real s. It is denoted by , where means the algebra of reals. We further postulate that in iff for all . As we do in polynomials, we shall agree that the terms not present indicate that the coefficients are understood to be zero. The elements of may be considered as elements of . In particular, the unity 1 of is also the unity of . Also, the element with belongs to , it being the formal power series with and all other ‘s zero. We now have the following important proposition which is the only tool necessary for working with formal power series as far as combinatorics is concerned:
Proposition : 2_4:
The element f of given by (2v) has an inverse in iff has an inverse in .
If is such that , the multiplication rule in tells us that so that is the inverse of . Hence, the “only if” part is proved.
To prove the “if” part, let have an inverse in . We will show that it is possible to find in such that . If such a g were to exist, then the following equations should hold in order that , that is,
So we have from the first equation. Substituting this value of in the second equation, we get in terms of the ‘s. And, so on, by the principle of mathematical induction, all the ‘s are uniquely determined. Thus, f is invertible in . QED.
Note that it is the above proposition which justifies the notation in , equalities such as
The above is true because the RHS has an inverse and
So, the unique inverse of is and vice versa. Hence, the expansion of as above. Similarly, we have
and many other such familiar expansions.
There is a differential operator in in , which behaves exactly like the differential operator of calculus.
Then, one can easily prove that is linear on , and further
from which we get the term “Taylor-MacLaurin” expansion
In the same manner, one can obtain, from , which in turn is equal to
the result which mimics the logarithmic differentiation of calculus, viz.,
The truth of this in is seen by multiplying the series on the RHS of (2vii) by the series for , and thus obtaining the series for .
Let us re-consider generating functions now. We saw that the GF for combinations of is .
Let us analyze this and find out why it works. After all, what is a combination of the symbols : ? It is the result of a decision process involving a sequence of independent decisions as we move down the list of the ‘s. The decisions are to be made on the following questions: Do we choose or not? Do we choose or not? Do we choose or not? And, if it is an r-combination that we want, we say “yes” to r of the questions above and say “no” to the remaining. The factor in the expression (2ii) is an algebraic indication of the combinatorial fact that there are only two mutually exclusive alternatives available for us as far as the symbol is concerned. Either we choose or not. Choosing “” corresponds to picking the term and choosing “not ” corresponds to picking the term 1. This correspondence is justified by the fact that in the formation of products in the expression of (2iv), each term in the expansion has only one contribution from and that is either or .
The product gives us terms corresponding to all possible choices of combinations of the symbols and — these are:
standing for the choice “not-” and “not-”
standing for the choice of and “not-”
standing for the choice of “not-” and .
standing for the choice of and .
This is, in some sense, the rationale for (2iv) being the OGF for the several r-combinations of .
We shall now complicate the situation a little bit. Let us ask for the combinations of the symbols with repetitions of each symbol allowed once more in the combinations.
To be discussed in the following article,
Combinatorics, Theory and Applications, V. Krishnamurthy, East-West Press.
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