PSU CS 106 Computing Fundamentals II Boolean Algebra - Intro HM 1/3/2009

2 © Dr. Herbert G. Mayer Agenda George Boole Augustus De Morgan Boolean Values 16 Boolean Operators Boolean And Boolean Or Boolean Xor Boolean Not De Morgan’s Theorem Useful Urls

3 © Dr. Herbert G. Mayer George Boole George Boole, 1815 to 1864, was a British mathematician, who developed what we now call Boolean Algebra (BA). He led a relatively uneventful life, being a private teacher at a young age in Doncaster, England, then math teacher in Liverpool, and in 1849 received an appointment as the first professor of mathematics at Queen's College, Cork in Ireland. He published numerous papers, many of which failed to create excitement in the world of mathematicians at the time. Of these, several are now regarded sine-qua-non standard material in Boolean Algebra, starting with his earliest publication on Theory of Analytical Transformations. Interestingly, the effective founder of the theory of computer science did not regard his specialty, BA, to be a branch of mathematics. His work influenced another contemporary founder of BA, Augustus De Morgan, living also in England from 1806 to 1871.

4 © Dr. Herbert G. Mayer Augustus De Morgan Augustus De Morgan was born in 1806 in Mandura, India. Lived in England, graduated from Trinity College, Cambridge in 1827, with well-rounded, classical education. Became professor of mathematics at University College in London. De Morgan was renowned for his methods of making math interesting and stimulating to students. Died 1871 in London after a life of heavy change with a vast body of publications. Well known as Boole’s contemporary and articulator of De Morgan’s theorem, though it had been known already in the times of Aristotle.

5 © Dr. Herbert G. Mayer Boolean Values The value domain of Boolean Algebra (BA) is limited to the two values True and False, in the computer science world also known as 0 and 1, represented via a bit (a binary digit) of information that can take only one of these values. In the computer technology such a bit is represented by defined voltage values. For example, a value close to 0 V stands for False (or 0), while an electrical value of 1.1 V stands for True (or 1). The real value of using bits comes in their combination to large sequences of bit strings. Boolean functions (operators) are logic operation on bits. There are 16 distinct operations, including the trivial identities of always true or always false; all 16 are shown below:

6 © Dr. Herbert G. Mayer 16 Boolean Operators Boolean Function F n Operator Symbol with at most two Boolean operands a and b Function Name F0F0 0Constant False F1F1 a ^ ba and b F2F2 a ^ ! ba and not b F3F3 aa identity F4F4 ! a and bnot a and b F5F5 bb identity F6F6 a x ba exclusive or b F7F7 a v ba or b – AKA inclusive or F8F8 a !v ba nor b F9F9 a x !v ba exclusive nor b F 10 ! bnot b F 11 a v ! ba or not b F 12 ! anot a F 13 ! a v bnot a or b F 14 a !^ ba nand b F 15 1Constant True

7 © Dr. Herbert G. Mayer Boolean And Boolean function and: true only if a and also b are true:

8 © Dr. Herbert G. Mayer Boolean Or (Inclusive Or) Boolean function inclusive or: true if a or b or both are true:

9 © Dr. Herbert G. Mayer Boolean Xor (Exclusive Or) Boolean function exclusive or: true if value of a differs from b:

10 © Dr. Herbert G. Mayer Boolean Not Boolean function not a: is true only if a is false:

11 © Dr. Herbert G. Mayer De Morgan’s Theorem Assuming a and b are Boolean variables And ^ the logical-and, and v the logical-or op Finally ! stands for the logical-not. Then: 1.) ! ( a ^ b ) = ( ! a ) v ( ! b ) 2.) ! ( a v b ) = (! a ) ^ ( ! b ) 1.) if the combined a and b are false, then at least one of a and b must be false 2.) if the combined a or b condition is false, then both a and also b must be false

12 © Dr. Herbert G. Mayer Useful urls