Q Suppose there're 25 students in your discussion Whichone is more likely LOO there're two students with the same birthday everyone has different birthdays A proof is a finite list of logical deductions which establishes the truth of a statement
1 Direct Proof Prove p Q Cta Assume P Method Deduce Q integers we say a divides b 15 Given a be 21 a 1 0 EI written Alb if F d C 21 ad b FPIPV a.bi.bz C 21 if alby and alby then at bi bz Let a bi ba C 21 Pf Assume alb a Iba and By definition F di de 21 such that ad adz be bi bi ad a Cd da Then ba adz a Ibi bae By definition D Reem To prove tx Ped in the domain and prove PCC pick an arbitrary C To prove Fx Poc one element d in the domain and show P d find
2 Proof by contraposition n p Recall the contrapositive of P Q nd is Geol Prove P Q 7 Q n p Method Prove E.ge prIPl Let me 21 Prove that if m2 is then is even even n attempt direct proof rt is even 2k F KEK n IFK n Pf Equivalent we'll prove n is odd rt is odd is odd Assume n 1 for some KE.TL A 2121 Then 2kt 1 2 4 k't 4kt I Thus n 2 2K't 2k t 1 is odd By definition n a
3 Proof by Contradiction total prove P Method Assume 7 P Deduce R M R for some R E.ge Def.I A real number r is rational if there exist p q Ez with 91 0 such that r a solution to Let 52 be TPropy 2 x Then 52 is irrational Assume 52 is rational PI 0 be such that 52 Iq Let P q EI q we can pick p of such that p q Notice that don't have common divisors By the definition of TI GEE PE p is even en 2K for some KEK By definition P 4k2 Then p2 2K 2q2_ Rbi q2 q2 is even is q even Contradict with p of having no common divisor 52 must be irrational b
E.ge Def.IAprime number is a natural number greater than 1 whose only positive divisors are 1 and itself ftp.T There are infinite many prime numbers Assume there are finitely many primes P 1 Pn Consider of Pn 11 P P I q Pn is not prime Since 9 f Pi for i L q n I a prime divisor of suppose p is q Since pl g thus p I g Pi Pn pl Pe Pn p I I ie contradiction Thus there are infinitely many primes 4 Miscellaneous we show p To prove equivalence i e Q Q p and Q P Oftentimes proof of equivalence is phrased as if and only if iff W LOG without loss of generality assumption means that follows is chosen armbitrarily ISSUE the a particular case narrowing down the statement to but doesnotaffectathevalidityoftheproofin general
If both Ky and ooty are x y C 21 E.ge TrTIp Let x and y then both are even even is odd or y is odd If either x contrapositive either Xy is odd or Xty is odd is odd WW CT assume 2M 11 i e X for 21 some M Sometimes it's useful to divide up proof into exhaustive cases want show either Xy is odd PI cont we or is odd Xty caseli.yi.se Then y for some n C 21 2n Hence Xty 11 2Mt It 2h 2 mtn is odd y is odd case 2 2n 11 for some n C 21 Then y Hence Xy 2Mt 1 Intl 4 m n t 2Mt 2h t I 2h2mn 1Mt n t 1 is odd This proves the statement by contraposition is
Constructive Existence Proofs Recall that to show F xp find an element a x such that Pla is true x y such irrational E g There exist that is rational is 2 Then KY eh Let Pf g Ln 2 X e 2 D Nonconstructive Existance Proof IT x y such here exist irrational that EI Q xD rational is P Assume 5252 is rational Pf nvm Then let X y and we're done Fz Fraisirional TP Assume 522 1525452 Then 2 is rational we're done D p Q a Pvp p a T since implication is correct and hypothesis we know the conclusion is correct is correct
QI 1wTSp Assume P Prove Q Goal I
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