By Jiri Herman, Radan Kucera, Jaromir Simsa
ISBN-10: 0387955526
ISBN-13: 9780387955520
This publication offers tools of fixing difficulties in 3 parts of uncomplicated combinatorial arithmetic: classical combinatorics, combinatorial mathematics, and combinatorial geometry. short theoretical discussions are instantly by way of conscientiously worked-out examples of accelerating levels of hassle and by way of routines that diversity from regimen to really demanding. The publication positive factors nearly 310 examples and 650 exercises.
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Example text
Therefore, the labeling of all ten flowers can be done in 5! -3! -2! different ways once the flowers are planted in any of A different ways. Therefore, A • 5! • 3! • 2! , or, in other words, This argument can easily be generalized to a general theorem. However, we will need a greater level of abstraction in our notations to achieve that. This is because we will take general variables for the number of objects, but also for the number of different kinds of objects. In other words, instead of saying that we have five red flowers, three yellow flowers, and two white flowers, we will allow flowers of k different colors, and we will say that there are a\ flowers of the first color, a-i flowers of the second color, 03 flowers of the third color, and so on.
Different ways. The two white flowers could be given two different labels in 2! different ways. Moreover, the labeling of flowers of different colors can be done independently of each other. Therefore, the labeling of all ten flowers can be done in 5! -3! -2! different ways once the flowers are planted in any of A different ways. Therefore, A • 5! • 3! • 2! , or, in other words, This argument can easily be generalized to a general theorem. However, we will need a greater level of abstraction in our notations to achieve that.
Equivalently, / is a bijection if for all y £ Y, there exists a unique x € X so that f(x) = y. In other words, a bijection matches the elements of X with the elements of Y, so that each element will have exactly one match. 9. Let / : X —> Y be a function. 8, then we say that / is one-to-one or injective, or is an injection. 8, then we say that / is onto or surjective, or is a surjection. 10. Let X and Y be two finite sets. If there exists a bijection f from X onto Y, then X and Y have the same number of elements.
Counting and Configurations by Jiri Herman, Radan Kucera, Jaromir Simsa
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