From bbd1df2901e03fb1da815502076d8c5c188cbade Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Sat, 18 Feb 2017 16:22:23 +0200 Subject: [PATCH] Corrections --- luku22.tex | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/luku22.tex b/luku22.tex index 2dcf6ab..25e177b 100644 --- a/luku22.tex +++ b/luku22.tex @@ -29,7 +29,7 @@ for the number $4$: A combinatorial problem can often be solved using a recursive function. In this problem, we can define a function $f(n)$ -that counts the number of representations for $n$. +that gives the number of representations for $n$. For example, $f(4)=8$ according to the above example. The values of the function can be recursively calculated as follows: @@ -42,7 +42,7 @@ can be recursively calculated as follows: The base case is $f(1)=1$, because there is only one way to represent the number 1. When $n>1$, we go through all ways to -select the last number in the sum. +choose the last number in the sum. For example, in when $n=4$, the sum can end with $+1$, $+2$ or $+3$. In addition, we also count the representation @@ -63,7 +63,7 @@ It turns out that the function also has a closed-form formula f(n)=2^{n-1}, \] which is based on the fact that there are $n-1$ -possible positions for +-signs in the sum, +possible positions for +-signs in the sum and we can choose any subset of them. \section{Binomial coefficients} @@ -369,8 +369,8 @@ The following rules precisely define all valid parenthesis expressions: \begin{itemize} -\item The expression \texttt{()} is valid. -\item If a expression $A$ is valid, +\item The empty expression is valid. +\item If an expression $A$ is valid, then also the expression \texttt{(}$A$\texttt{)} is valid. \item If expressions $A$ and $B$ are valid, @@ -397,7 +397,7 @@ such that both parts are valid expressions and the first part is as short as possible but not empty. For any $i$, the first part contains $i+1$ pairs -of parentheses, and the number of expressions +of parentheses and the number of expressions is the product of the following values: \begin{itemize} @@ -559,12 +559,12 @@ that corresponds to the area of the region that belongs to at least one circle. The picture shows that we can calculate the area of $A \cup B$ by first summing the -areas of $A$ and $B$, and then subtracting +areas of $A$ and $B$ and then subtracting the area of $A \cap B$. The same idea can be applied when the number of sets is larger. -When there are three sets, the inclusio-exclusion formula is +When there are three sets, the inclusion-exclusion formula is \[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \] and the corresponding picture is @@ -594,7 +594,7 @@ If the intersection contains an odd number of sets, its size is added to the answer, and otherwise its size is subtracted from the answer. -Note that similar formulas can also be used +Note that there are similar formulas for calculating the size of an intersection from the sizes of unions. For example, @@ -678,10 +678,10 @@ elements should be changed. \index{Burnside's lemma} -\key{Burnside's lemma} counts the number of -combinations so that +\key{Burnside's lemma} can be used to count +the number of combinations so that only one representative is counted -for each group of symmetric combinations, +for each group of symmetric combinations. Burnside's lemma states that the number of combinations is \[\sum_{k=1}^n \frac{c(k)}{n},\]