Discrete Mathematics with Graph Theory (3rd Edition) 28

Discrete Mathematics with Graph Theory (3rd Edition) 28 -...

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26 Solutions to Exercises p: I like mathematics (I) [BB] Let p, q, and r be the statements ; ~ ~ ~:~y mathematics The given argument is This is the same as p~q (-,q) V r (-'8) ~ (-,r) 8~q 8~q 8 : I graduate. which is certainly not valid, as the fOllowing partial truth table shows. p: I like mathematics (m) Let p, q, and r be the statements q: I study r : I pass mathematics 8 : I graduate. p~q (-,q) V r . (-'8) ~ (-,r) The given argument is which is the same as p~8 which is valid by two applications of the chain rule. 6. [BB] r V q is logically equivalent to [-,( -,r) V qj <=> [( -,r) ~ qj so, with p ~ -,r, we get p ~ q by the chain rule. 7. We will prove by contradiction that no such conclusion is possible. Say to the contrary that there is
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Unformatted text preview: such a conclusion e. Since e is not a tautology, some set of truth values for p and q must make e false. But if r is true, then both the premises (-,p) ~ rand r V q are true regardless of the values of p and q. This contradicts e being a valid conclusion for this argument. 8. (a) [BB] p 1\ q is true precisely when p and q are both true. (b) By 8(a), we can replace pl\qby the twopremisesp and q. Using modus ponens, p andp ~ r lead to the conclusion r. Using modus tollens, q and 8 ~ (-,q)) lead to the conclusion -'8. Finally, 8(a) says we can replace -'8 and r with (-'8) I\r. 9. By Exercise 8(a), the final premise is equivalent to the list of premises q1. q2, ... ,qn. Now...
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