(b) Exercise 3 of Section 1.6 (c) Theorem (d) The PMI, Section 2.4 (e) Theorem (0 Theorem (g) Theorem4.2.2

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3 16 (HAPTER 1 logic and Proofs ). 3. (h) (i) The fish bite only when the moon is full, A time of 3 minutes, 48 seconds or less is necessary to qualify for the Olympic team. Write the converse and contrapositive of each conditional sentence in Exercise 1. What can be said about the truth value of Q when (a) PisfalseandP* Q istrue? (b) PistrueandP* Qistrue? (c) PistrueandP+ Q isfalse? (d) PisfalseandP+) Qistrue? (e) P is true and P f Q is false? 4. Identify the antecedent and consequent for each conditional sentence in the following statements from this book. (a) Theorem 1.3.1(a) (b) Exercise 3 of Section 1.6 (c) Theorem (d) The PMI, Section 2.4 (e) Theorem (0 Theorem (g) Theorem4.2.2 (h) Theorem 5.1.7(a) * 6. * Which of the following conditional sentences are true? (a) Iftriangles have three sides, then squares have four sides. (b) If a hexagon has six sides, then the moon is made of cheese. (c) It7 +6:14,then5 *5:10. (d) If 5 <2,then 10 < 7. (e) If one interior angle of a right triangle is 92", then the other interior angle is 88o. (f) If Euclid's birtlday was April 2, then rectangles have four sides. (g) 5 is prime if 42 is not irrational. (h) 1 + 1:2issufficientfor3 > 6. Which of the following are true? (a) Triangles have three sides iff squares have four sides. (b) 7 +5:12iff I1-l:2. (c) b is even iff b + I is odd. (Assume that b is some fixed integer.) (d) n is odd iff m2 is odd. (Assume that m is some fixed integer.) (e) 5*6:6+5iff7*l:10. (f) A parallelogram has three sides iff 27 is prime. (g) The Eiffel Tower is in Paris if and only if the chemical symbol for helium is H. (h) "/ro + Jn. Ji + Jnn JB - Jn.. Jn - ho. (i) xz > O iff -r > 0. (Assume that x is a fixed real number.) 0) *2 - y': 0 iff (x - y)(x * y) : 0. (Assume that.r and y are fixed real numbers.) (k) xz+y2:50iff (x-ty)2:50. (Assume that x and y are fixed real numbers.) Make truth tables for these propositional ",7-i forms. (a) P+(QnP). ; (c) -Q+ (Q e P). * (b) (-P + e)v (e <+ p). - (d) (PvQ)+e^Q). (e) (PnQ)v(Q^R)=+PvR. (f) l(q+ s) ^ (0 + R)l + te v Q) + (s v R)1.

4 1.2 Gonditionals and Biconditionals Prove Theore m I.2.2by constructing truth tables for each equivalence' g, Determine whether each statement qualifies as a definition' (a) y : f (x)is a linear function when its qranh fs a totgl:llt 'y iu : f (x) is a quadratic function when it contarns an x- term' 'm i"t is apetfect'iquare when m n2 fyr some integer n', (d) A triangle it iigttt ttl"ngle when the sum of two of its interior angles " is 90'. (e) Two lines are parallel when their slopes are the same number' (fi A sundial is an instrument for measuring time',rfi, R"rrite each of the following sentences using logical connectives. Assume '; ^'"',rt"i symbol/, xo, n, x,s' B represents some fixed object' *(a)iflhasu,"lutiu"minimumatxsandif/isdifferentiableat'x0'then """rt //(x6) : 0' ** (b) if n is prime, then n :2 or zr is odd' (c) A numberu is and not rational whenever x is irrational' 'eul * (d) If x: 1 ot x: -1, then lx : 1' * i"j /has a critical point at xs\ff f '(xs): 0 or/'(xs) does not exist' (f) S is compact iff S is closed and bounded' (g) B is invertible is a necessary and sufficient condition for det B + O' irtl 6>-n - 3 only if n> 4or n> I0' (i) r is Cauchy implies x is convergent' 0) /is continuous at x6 whenever f (x) : f (x$',l$ (k) If/is difl.erentiable ar xe and/is strictty increasing at rs, then/'(.ro) > 0. ll.dictionariesindicatethattheconditionalmeaningofunlessispreferred,but thereareotherinterpretationsasaconverseorabiconditional.discussthe translation of each sentence' (a) I will go to the store unless it is raining' *(b)thedolphinswillnotmaketheplayoffsunlessthebearswinalltherest of their games' (c) You cannot go to the game unless you do your homework first' idl You won't win the lottery unless you buy a ticket' fd Show that the following pairs of statements are equivalent' \-/*-(a) (P v 0) + R and -R J (-P r'-q)' * inl e rq)+rand(pn-r)+-q' -- i.) P + (Q n R) and (-Qv -R) + -P' (d) P+(QvR)and (P ^-R)+Q' G) e+q)irand(pn-q)vr' (f) P eqand(-p v Q)^eQv P)' 13. Give, if possible, an example of a true conditional sentence for which * (a) the converse is true' ^ (b) the converse is false' * (c) the contrafositive is false' (d) the contrapositive is true' 14. Give, if possible, an example of a false conditional sentence for which (a) the converse is true' (b) the converse is.false' (c) the contrapositive is false' (d) the contrapositive is true'

5 l8 CHAPTER I Logic and proofs 15' Give the converse and contrapositive ofeach sentence ofexercises (c), and (d). 10(a), (b), Tell whether each converse and contrapositive is true or f-alse. \ 16' Determine whether each of the following is a tautology, * a confradiction, (a) or [(P=+e)+4ap. neither. F (b) P++pn(PvQ). (c) P+eepn-e. * (d) P =+ [p =+ (p + e)1. (e) P n (Qv -e) ++ p. (f) tq^e+e)l=+p. (g) (P +) g ++ -(-p v e)v (-p q. (h) tp + (QvR)l =+ le+ ilv (R+p)1. ^ (i) pn(pee)n_e. CI) ev e)+e+p. (k) tp+(e^r)l+ tr=+ (p +Dl (l) tp+(e^r)l +R+ (p*a'.' 17. The inverse, or opposite, of the conditional sentence p =+ eis _p (a) + _e. Show that p + e and its inverse are not equivalent (b) forms. For what values ofthe propositions p and e are p ) e andits both true? inverse (c) which is equivalent to the converse of a conditional sentence, trapositive the con_ ofits inverse, or the inverse ofits contrapositiufi