19 Boolean Algebra

19 Boolean Algebra - Chapter 19 out of 37 from Discrete...

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Chapter 19 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 1 After George Boole (1815-1864). 1 19 Boolean and Set Algebra It has been known for sometime in mathematics that set algebras and Boolean 1 algebras are different perspectives on the same thing. The treatment of sets here is informal and is known as naive set theory . Most of the time naive set theory is sufficient for the purposes of even professional mathematicians. Those familiar with this will want to skip the first part of the chapter. However, later there is some useful material not usually found in texts. Set Algebras Boolean Algebras A set is a collection of objects. This is not a formal definition but a casual definition. A formal definition of sets is deceptively difficult and is unnecessary for our purposes. A Boolean algebra is a logic algebra. The variables take on two values corre- sponding to truth (1 or T) and false (0 or F).
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Chapter 19 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 2 The Universal set can be denoted by U or by 1. Figure 1 The Universal Set Truth is denoted by T or by the integer 1. The empty set is denoted by i or 0. The empty set is sometimes known as the null set . (Think of an empty file cabinet, or equivalently a college administrator's mind.) False is denoted by F or by the integer 0. Figure 2 The Empty Set
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Chapter 19 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 3 Figure 4 The Complement of the Set X A set is a collection of objects. (In this case it is represented by the interior of a circle.) Figure 3 A Set X A Boolean variable X can take on either of two values 1 (T) or 0 (F). The complement of a set is the collection of all objects not in that set. (If the set is the interior of the circle, the complement is the outside). The complement of set X is denoted by X ) or by X'. Not X is denoted by - X. If X = 1 then - X = 0. If X = 0 then - X = 1.
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Chapter 19 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 4 Figure 5 A c B Figure 6 A 1 B The union of set A and set B is denoted A c B (sometimes A + B). A or B is denoted by A w B (sometimes A + B). A = 1 and B = 1 Y (implies) A w B = 1. A = 1 and B = 0 Y A w B = 1. A = 0 and B = 1 Y A w B = 1. A = 0 and B = 0 Y A w B = 0. The intersection of set A and set B is denoted A 1 B (sometimes A A B). A and B is denoted by A v B (sometimes A A B). A = 1 and B = 1 Y A v B = 1. A = 1 and B = 0 Y A v B = 0. A = 0 and B = 1 Y A v B = 0. A = 0 and B = 0 Y A v B = 0.
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Chapter 19 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 5 Figure 7 A ! B.
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This note was uploaded on 01/19/2012 for the course ENG 151 taught by Professor Atkins during the Fall '10 term at University of Michigan.

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19 Boolean Algebra - Chapter 19 out of 37 from Discrete...

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