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L21_ConverseAndContrapositive-1 - Negating an Implication...

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Unformatted text preview: Negating an Implication Negating We have seen that ~(pVq) is logically equivalent to ~pΛ~q . And we have seen that ~(p Λq) is logically equivalent to ~p V ~q . Question: How do you negate p => q? You might think that the negation of an implication is another implication, but it isn’t! implication In fact, since “p =>q” is logically equivalent to In the OR statement “~p V q” , the negation “~(p=>q)” is logically equivalent to the negation “~( ~p V q)” . This is the same as “~~p Λ ~q” . ~( The double negation ~(~p) is just p. Thus “~(p=>q)” is logically equivalent to “p Λ ~q”. We have just learned that the negation of an implication is an AND statement. implication Example: Negate the statement “If it will rain I Example: will go to the movies.” will Here p: It will rain and q: I will go to the movies. The negation of “p=>q” is “p Λ ~q”, which is: The “It will rain and I will not go to the movies.” Example: Negate the sentence: Studying daily implies you will get good grades. implies Let p: (You) will study daily q: You will get good grades. q: The given sentence is p=>q. The Its negation is p Λ ~q , which is the sentence: “You will study daily and you will not get good grades.” Converse and Contrapositive Converse In this lesson we learn two ways to create In a new implication from a given one. new Given a statement “If p then q” the converse is the statement “If q then p” converse obtained by switching the positions of p and q. and Example: Let p and q be the statements p: I have brown hair p: q: I have brown eyes. q: The implication “If p then q” is the statement The A: “If I have brown hair then I have brown eyes.” The converse is the statement B: “If I have brown eyes then I have brown hair.” We will see that the truth or falsity of A has little We connection to the truth of falsity of B. connection Here are the truth tables for A and for B: Here (A) (B) p T T F F q T F T F if p then q T F T T if q then p T T F T The last two columns are different, so the truth The table for A is not the same as that for B. table It happens very frequently in ordinary speech as well as in mathematics that people confuse an implication with its converse. Try to be aware of this common but serious error so you don’t make it! make The contrapositive of the implication A: “If p then q” is the statement C: “If ~q then ~p” C: To obtain the contrapositive of A, negate both p and q and then reverse their positions. Let p, q be the statements from before: Let p: I have brown hair. q: I have brown eyes. A: If I have brown hair then I have brown eyes. The contrapositive of A is: The B: If I don’t have brown eyes, then I don’t have B: The contrapositive sounds complicated. Its importance comes from the fact that the contrapositive is logically equivalent to the original implication! One is true precisely when the other is true. Here are the truth tables. the (A) (B) (A) p q ~q ~p p 0q ~q0~p ~q T T F F T T T F T F F F F T F T T T F F T T T T We see that the last two columns are identical, which says that “p0q” and its contrapositive which “~q0~p” have the same truth values. In other ~p” words, they are logically equivalent, although they sound quite different. they Observe: the converse of the converse is the Observe: original implication. And the contrapositive of the contrapositive is the original implication. the ...
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