MA112-Lecture_Notes-OH-L03

MA112-Lecture_Notes-OH-L03 - MA112 Discrete Mathematics...

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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 1 MA112 Discrete Mathematics Omar Hamdy omar.hamdy@gmail.com Lecture Notes 3
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 2 Review: Logical Connectives iff Biconditional p?q:true if,then Conditional xor Exclusive or || or Disjunction and Conjunction ! not Negation Java Usage Symbol Operator
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 3 Constructing the Truth Table • First we count propositional variables (input) to determine number of possibilities (number of rows) = 2 n • We simplify the compound proposition by producing T.T. for the sub-propositions. • We observe precedence rules.
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 4 Tautology and Contradiction • A compound proposition that is always true for all the truth values of its propositions is called tautology. • The opposite of a tautology is a compound proposition that is always false. This is called contradiction. • A compound proposition that is neither a tautology nor a contradiction is called contingency.
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 5 Logical Equivalence • The propositions p and q are called logically equivalent if p q is a tautology. In other words if they have the same truth table. The notations p q or p Ξ q denote p and q are logically equivalent. • Equivalent propositions are important because they help making logical arguments and help simplify logical statements.
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MA112 Discrete Mathematics Dr. Omar Hamdy Summer 2010 6 Important Equivalences • Identity Laws • Domination Laws • Idempotent Laws • Negation Laws • Double Negation Law • Commutative Laws • Associative Laws • Distributive Laws • De Morgan Laws
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Dr. Omar Hamdy Summer 2010 7 Predicate Logic • Propositional Logic Limitation Example: – If falcons are birds, then falcons lay eggs – p = falcons are birds, q = falcons lay eggs – p q – If we know that eagles are birds, can we also imply they lay eggs from the previous? • Propositions do not have centric
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MA112-Lecture_Notes-OH-L03 - MA112 Discrete Mathematics...

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