Proofs-07-InClass-Mar1017-FilledIn.pdf

Proofs-07-InClass-Mar1017-FilledIn.pdf - comp2130 W17...

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Unformatted text preview: comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Won- lO/ l 7’ Proof Problems: 1-34; Definitions: 1-4; Theorems 1-3; H C ‘ - “ {‘57 "WM, 915 — no»; a .I 6,, Hum-.0 “MA”! —_——__*_¢ Prove 26: There exist rational number r such that for every irrational nu the number r*s is rational Proof: Follow a similar strategy suggested in the previous proof problem. Let ::_ a; 7: :1. ways lrrw‘m’K “no- J?- wa; 0X5:D,YM Then, for each irrational number s, it follows that I Recall that the negation of VX E 5’ 33’ E T,R(x,y) is . ~ (Vx E S, Ely E T,R(x,y)) a (3x E S, Vy E T, ~ R(x,y)) . Therefore, to show that quantified statement 46 comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1-4; Theorems 1-3; =44; is #, we must show that * i195 such that V fiT 201,11) is false. Disprove 27: the ollowing statement: ' te er 11, there exists a negative inte er m such that n+m=1 Proof: + .- The statement in symbols is: 37L W E Z I 3 ”16-2 9- mm :31. Note we have to M this statement. - .1- That 1s, __ + N (heal: me; ,2n+m=1 "’52 i. e. ,we will prove the righ hand side. n+m:): _1_ £115ng gummy") positive integer. Consider Then for every negative integer m, (Ll-m; 1+?” g_ ’lfl-U 5Q i=1 ’l . LW+W> 47 comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1—4; Theorems 1-3; Therefore, *7: l is a wygample. Disprove 28: th following statement: ' s a positive integer n such that for every negative integer m, n+m <0" Proof: We show that M 'w M Q' HM MW V’ i Let L: be a positive integer WW3 Then m : —fik is a negative integer and Prove 29: There is no smallest positive real number May. I0, 17 go-Faz. Pave—d unimmo w-H-Pe-t Wmné- Free-F Of \CMHM '9an by Gum \JJI’fiauJ- G-F Lax anfi, OW Prob-van»: .L—lS' 13’4”“; UNVMAaO amt-ppm Ros/cal ZxWHflQWW 5mm? W 19-23 Ddst/d Gael-emu 7WW Gl-uhm Probing 24- %d @MWW Bahama 2-S— 24 19.1mm 69mm ‘ZWW Ahtmtg Prawn-m: 25‘— 23 480k u I i I l : ! E I 31 ’i ¥xeg (2(1) Dug“ PM,” (at) am?» W (962) «'4 W , bun/acvv‘: 'hue. QILraPDS-v‘hx. 3" DC €31 Q“) P002 42(1) Mala) :=> N POL) W ,‘ N&-C1) Wu. «gm ' 3'4 W3 {95°sz Rae» warm 48b comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1—4; Theorems 1-3; Proof: Restate the problem 1n symbols: #265: £952 9— 90¢ We have investigated the truth or falseness of quantified statements of the type VXES, R(X), where R(X) is an implication P(X) => Q(X) over a domain S. Direct Proof: (otherwise: MM Assume: P190 1”) M Show that: Lima (2 fig Proof by Contrapositive; (otherwise: M0496 TDi'eM __) 49 comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1-4; Theorems 1-3; Assume: Qg z; 2 Age Show that: 2‘ z 2 Q 19%: What if we cannot use these two methods to prove a given statement of the form Vxe S, R(X) ? Assume RC1.) V.) 341?; ° For quantified statements R such as W? =9RC7L) _ with open sentences P(x) and Q(x) over domain S, we have already described two methods that we might use to verify the truth of the statement, namely, Direct Proof and Proof by Contrapositive. YA ° Let us now see a 3 method assume (N B.) to dedu asta ement that ”(WM :ésoe Wm 43% we made 1n the proof. mm 50 comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1-4; Theorems 1-3; ° If we denote the fact or assumption by L, then what we have deduced is NE, . ° That means we have produced a which we denote by _c A N ) ° That is, we have shown that: l<A~K is C ° In other words, we have the k H_J_LH\ VM of the implication NR 3‘3 as M. ° However since L is 4% and NR =$ C is m, it follows (by NR=>Qlwlww that fig is :M and therefore, R. is +YuQ., the desired outcome. 51 comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1-4; Theorems 1-3; To prove that a statement R is true using a Proof by Malaria W6 assume that mm from this, obtain a Muslim If R is expressed in terms of an implication, say R: V XE S, P(x)=> Q(x), then we begin a Proof of Mum by assuming that R is .419 i.e., by assuming that V XE S, P(x)=> Q(X) is _-_Pq_h¢ We have seen that this means there exists some element xES for which P(x) is homand Q(x) is ’Qfi We then attempt to produce a M for this. Prove 29 gagain): There is no smallest positive real number Proof: Restate the problem in symbols: Vane—12+; ave-12* 9 9424 52 comp2130 W17 Proofs — Proof Methods Proof; Proof of universal quantifier; Odd and Even Numbers, Direct Proof; Proof of binconditional; Proof by Contrapositive; Proof by cases; Omitting proof without loss of generality; disproving (counterexample); Proof of existential quantifier; Proof by contradiction; Proof Problems: 1-34; Definitions: 1-4; Theorems 1-3; Mafia W ‘m Mmmlm 0» WWW 9 W W Wm WM red W %' >0 \‘e o<§<aH€ L’fambaww W 44¢:qu numbufla. mwm‘fgw Observe: In Proof by Contrapositive, we arrived at a Prove 30: No even integer can be expressed as the sum of an even integer and an odd integer . . . . (A) Proof: Suppose that the statement is 53 ...
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