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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: 134; Deﬁnitions: 14; Theorems 13; 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: 134; Deﬁnitions: 14; Theorems 13; =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 ”162
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, (Llm; 1+?” g_ ’lﬂU 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: 134; Deﬁnitions: 1—4; Theorems 13; 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 goFaz. Pave—d unimmo wHPet Wmné
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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: 134; Deﬁnitions: 1—4; Theorems 13; 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: 134; Deﬁnitions: 14; Theorems 13; 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: 134; Definitions: 14; Theorems 13; ° 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: 134; Deﬁnitions: 14; Theorems 13; 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+; ave12* 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: 134; Deﬁnitions: 14; Theorems 13; 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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 Summer '14
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