Abstract
Verify the following relations involving the operations ∩ (intersection) and ∪(union): A ∪B = B ∪A, A ∩ B = B ∩ A (commutativity), A ∪(B ∪C) = (A ∪B) ∪C, A ∩ (B ∩ C) = (A ∩ B) ∩ C (associativity),A ∩ (B ∪C) = (A ∩ B) ∪(A ∩ C), A ∪(B ∩ C) = (A ∪B) ∩ (A ∪C) (distributivity), A ∪A = A, A ∩ A = A (idempotent property of ∩ and ∪).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The definitions and some basic facts concerning the Stirling numbers (of the first and the second kind), and also of the Bell numbers, can be found in Sect. A.1.
- 2.
Tradtionally linked to the population growth of a colony of rabbits, and described as early as the thirteenth century AD, by Leonardus Pisanus de filiis Bonaccii, widely known under the nickname “Fibonacci,” in his book “Liber Abaci,” probably written around 1202 CE.
References
Berger, M.A.: An Introduction to Probability and Stochastic Processes. Springer, New York (1993)
Bol′shev, L.N., Smirnov, N.V.: Tablicy matematicheskoĭ statistiki. Nauka (Science), Moscow (1983)
Feller, W.: An Introduction to Probability Theory and Its Applications, vols. I and II, 3rd edn. Wiley, New York (1968)
Fristedt, B., Gray, L.: A Modern Approach to Probability Theory. Birkhäuser, Boston (1997)
Khrennikov, A.: Interpretations of Probability. VSP, Utrecht (1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Shiryaev, A.N. (2012). Elementary Probability Theory. In: Problems in Probability. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3688-1_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-3688-1_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-3687-4
Online ISBN: 978-1-4614-3688-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)