# P p p ν p p λ p t f f t t t f f 60 definition 1 a

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p¬ppν¬ppΛ¬pTFFTTTFF60DEFINITION 1A compound proposition that is always true, nomatter what the truth values of the propositionsthat occurs in it, is called atautology.A compound proposition that is always false iscalled acontradiction.Acompoundpropositionthatisneitheratautologyoracontradictioniscalledacontingency.
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Propositional EquivalencesCompound propositions that have the same truth values inall possible cases are calledlogically equivalent.Example: Show that ¬pνqandpqare logicallyequivalent.Truth Tables for¬pνqandpq.pq¬p¬pνqpqTTFFTFTFFFTTTFTTTFTT63DEFINITION 2The compound propositionspandqare calledlogicallyequivalentifpqis a tautology. The notationpqdenotesthatpandqare logically equivalent.
Propositional EquivalencesIn general, 2nrows are required if a compound propositioninvolvesnpropositional variables in order to get thecombination of all truth values.64
Propositional Equivalences65
Applications: Boolean SearchesLogical connectives are used extensively insearches of large collections of information.Example: indexes of Web pages.AND - used to match records that contain both oftwo search terms.OR - used to match one or both of two search terms.NOT - used to exclude a particular search term.Read about: Web Page Searching
Applications: Logic PuzzlesPuzzles (important job interview question) thatcan be solved using logical reasoning[Sm78] Smullyan: An island that has two kindsof inhabitants.knights, who always tell the truth.knaves, who always lie.You encounter two people A and B.What are A and B if:A says “B is a knight” andB says “The two of us are opposite types?”
Example 1:p: A is a knight ¬p: A is a knaveq: B is a knight¬q: B is a knaveConsider the possibility that A is a knight;So, p is true. And he is telling truth.So, q is true. So, A and B are the same type.However, if B is a knight, then B ’s statement that A andB are of opposite types, the statement (p ∧¬q) ∨ (¬p ∧q), would have to be true, which it is not, because Aand B are both knights. Consequently, we can concludethat A is not a knight, that is, that p is false.
Example 1: Solution (cont..)Consider the possibility that A is aknave,everything a knave says is false; q is true,is a lie.So, q is false. B is also a knave.B ’s statement that A and B are oppositetypes is a lie, which is consistent withboth A and B being knaves.We can conclude that both Aand B are knaves.
A father tells his two children, a boy and a girl, toplay in their backyard without getting dirty.However, while playing, both children get mud ontheir foreheads. When the children stop playing, thefather says “At least one of you has a muddyforehead,” and then asks the children to answer“Yes” or “No” to the question: “Do you know whetheryou have a muddy forehead?”The father asks this question twice. What will thechildren answer each time this question is asked,

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