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# Truth Tables, Logical Equivalence, Operator Associativity and Conditional Propositions CSC 245

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This page Class Notes was uploaded by Jane Onyishi on Monday February 8, 2016. The Class Notes belongs to CSC 245 at University of Arizona taught by Dr Lester McCann in Spring 2016. Since its upload, it has received 20 views. For similar materials see Discrete Mathematics in ComputerScienence at University of Arizona.

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Date Created: 02/08/16

Truth tables should include all the possible combinations of truth values for all the simple propositions involved To structure your truth table Alternate the right most column T F T F T F Alternate the next column to the left T T F F T T Alternate the next column to the left TTTT F F F F until you39re done hopefully you39re done before you get here Associativity of Operators All the previously discussed operators are leftassociative the left side of the operator gets evaluated rst except negation Use parenthesis to change the precedence of an operator because there are no concrete rules regarding the precedence of logical operators Logical Equivalence Two propositions are logically equivalent p E q if they produce the same output evaluate to the same truth values given the same input Three categories of propositions based on truth values Tautology Contradiction and ConUngency A tautology is a proposition that is always true A contradiction is a proposition that is always false and a contingency is a proposition that is neither always true nor always false neither a tautology nor a contradiction Example of a tautology T V p This proposition is always true because inclusive or is true when either side or both sides are true Example of a contradiction F A p This proposition is always false because a conjunction is only true when both sides are true Example of a contingency p Simplest contingency because p could either be true or false Conditional Propositions A proposition that can be written in the form quotif p then qquot where p and q are propositions is a conditional proposition Conditionals are written like so quotp gt qquot which is read as quotp implies qquot There are many English representations of implication some of which are if p then q p is suf cient for q if p q p only if q q unless p q if p q whenever p q is necessary for p NQP PWF Example of a conditional proposition If a proposition can be written in the form quotif p then qquot then it is a conditional proposition In this example p is quotA proposition can be written in the form quotif p then qquotquot and q is quotA proposition is a conditional propositionquot quotp gt qquot can be written as quotA proposition can be written in the form quotif p then qquot only if it is a conditional propositionquot or as quotA proposition is a conditional H H proposition unless it cannot be written in the form quotif p then q A conditional proposition is true p q is true unless p is true and q is false lnverse of an implication The inverse of p gt q is p gt uq Converse of an implication The converse of p gt q is q gt p Contrapositive of an implication Conversing the inverse or inversing the converse the contrapositive p gt q of is q gt up An implication is logically equivalent to its contrapositive Biconditional Propositions This is read quotp if and only if qquot and written in logic as quotp lt gt qquot This means that p implies q and q implies q p gt q A q gt p Example An object is a pencil if and only if it can make marks than can make erasable marks on paper p An object is a pencil q An object can make erasable marks on paper The implication can be written in logic as quotp lt gt qquot Two propositions p and q are logically equivalent if p lt gt q is a tautology De Morgan39s Laws These are very useful logical equivalences involving the negation of a compound proposition involving an AND or an OR a w5wvw wsw w

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