BooleanLogic

# BooleanLogic - CSE 1400 Applied Discrete Mathematics Boolean Logic Department of Computer Sciences College of Engineering Florida Tech Fall 2011

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Unformatted text preview: CSE 1400 Applied Discrete Mathematics Boolean Logic Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Control Structures 1 Boolean Logic 3 Normal Forms 4 The Conditional Operator 5 Complex Propositions 6 Truth Tables 8 Logical Equivalence 10 Satisfiable and Valid Propositions 11 Constructing Boolean Functions 12 Satisfiability 13 Counting Boolean Expressions 14 Consistency and Completeness of Boolean Algebra 14 Problems on Boolean Logic 15 Abstract Logic controls the action of a computer. Control Structures T he B öhm-J acopini theorem shows that only three funda- mental control structures are necessary to implement any T uring algorithm . 1 . Execute instructions sequentially one after another as in figure 1 2 . Select one of two instruction to execute according to the value of a Boolean variable as in figure 2 3 . Iterate through a sequence of instructions until a Boolean variable changes value as in figure 3 cse 1400 applied discrete mathematics boolean logic 2 Statement Next Statement Figure 1 : Sequential execution of statements. Boolean Condition Then Statement(s) Else Statement(s) Figure 2 : Conditional execution of statements. Boolean Condition Loop Statement(s) Figure 3 : Loop (iterative) execution of statements. cse 1400 applied discrete mathematics boolean logic 3 Boolean Logic B oolean logic provides the basis to control the execution of algorithms . P ropositional calculus studies the behav- ior of formulas constructed using Boolean variables. The domain Boolean variables are typically named p , q , r , s , . . .. of these variables is the set of truth values B = { False , True } . A Boolean variable is also called a proposition and is often inter- preted as the name for a declarative natural language sentence: A proposition is a sentence that can only be True or False . A formal Example propositions: • All humans are mortal. ( True ) • Socrates was human. ( True ) • Socrates was mortal. ( True ) • George Washington was President of the United States of America from 1789 to 1797. ( True ) • May 1 , 1911 was a Wednesday. ( False ) • 2 + 3 = 7 ( False ) • π = 3.14 ( False ) • All relations are functions. ( False ) syntax describes what propositions are and how they can be ma- nipulated. 1 . False , also called 0, is a proposition. 2 . True , also called 1, is a proposition. 3 . Boolean variables over the set of bits are propositions. If p is a Boolean variable, then p ∈ { 0, 1 } . 4 . If p and q are propositions, then (a) N ot p , denoted ¬ p is a proposition p and ¬ p are called literals . One of them is False , the other is True . (b) p and q , denoted p ∧ q is a proposition Haskell uses && to express and ....
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BooleanLogic - CSE 1400 Applied Discrete Mathematics Boolean Logic Department of Computer Sciences College of Engineering Florida Tech Fall 2011

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