sum ! _Desca

John Rachlin discrete structures

Northeastern

Cogito, ergo

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a logical formula from

truth table ,

F F T y¥#(CN FTF

include these cases TFTT )

f-a n – bnc)v1- anbnc)✓(an – bn- c)✓Can- bnc)

✓ (an b n – c) v Canbn e )

” Disguctiue

form ‘ (DNF):

(X. ^ ✗ z n . . . . )DM (g.n ya ^ . . .. )NBA ( – – – . )

Alternatively ,

The formula iwhatever

w e focus o n the false entries .

when – (- ar – b r – c) A – (- an b n –

De Morgan’s

: -(png)=-pv-q

so : n (na n – b n – c) = a ✓ b v c

v ^(narbr-c)=av-b C

(avbvc)rlav – b vc) “Conjunctive Normal Form ” (

(✗ ✓ Xzv. – – ,

) AAA ( ) MEH

(avbvc)A lav- b vc)

(ave)✓(bn- b)

(a v c Fave

Wa t c h F o r

These simplifying opportunities:

DNF • • . (pnqnr)✓(pngr- r)•*

^ (r ✓ – r ) distributive

complement

CNF•• • (pvqvr)^(pvqv-r)

Distributive

complement

O=F , 1=-1

LogicGates_

Arg=D- A^BAND

=☐-AVB OR A I)Io- NOR ,

A – Do- -A NOT%

÷i@ *§÷: .

±÷÷t÷EY÷Y÷:÷

N O T A N D N A N D O R N O R ✗O R ✗ N O R

-Do- =D- a.Do- =D- – =D- –

simplifyingacircu.la#-2D-DoBc-Ao-D-5

– ✗AnB)^ (Av- c))

((A^B^A)v(AnBn- c)

) Distributive

– ((AnB)✓ (A^B^- a)) AnA

Absorption

a n d humid

Designingacircuit-Anexan.rs/e-

Suppose 3 inputs Az A , Ao

which w e interpret a s a n unsigned binary #

111 or0to7,

lll ÷ Prime

Circuit outputs I if the input

(3-bits max)

Prime#: 2,3,5,7,11,13,17,19, s-

Note : • A prime # is a whole # > 1 whose

factors are 1 and itself.

• By convention , 1 is riot prime .

Numbers with more than two

factors a re

composite ”

• Largest known prim e has

23 million digits

• R S A Encryption

1 Oll l l00

f-azna n – ao)v1- azra nao)v(az^- a. ^ao)v(azra,n9o)

f-arena , )r 1-aovao)] ✓ [(aznao)^l- a. va ,)]

((-92^9)^-1) v ((aznao)1T )

f-az na , ) v (are ^ ao)

ao – D2 out ,

a .IT#-D- a

propositional

Aristotleofferedlhefollowing.by/logisim-

Socrates is

statement :

( ✗ is mortal ) But “X is a man” has no inherent

is a man)→

It’s true for som e Xs

and false for other.

firs-orderl.co#

claims about

X = representing

particular

predicates

establish claims about those

Human (Socrates) is tru e

Human (Socrates) → Mortal (Socrates)

universal Quantifier

th Human(x) → Mortal(x)

true for all objects , X

Prime(X)→ Odd(X)

False: ✗= 2 (one counter

example is !) enough

The existential Quantifier

” There exists In ✗

F. ✗ NEU – Professor (X ) ^ Teachers 1800 (X )

be true for

orieormore

Negatingf×_

– thPix)= Fx

[the Prime(x)→ oddCx)]

3-✗ a (Prime (x) → Odd (X) )

1-✗- l-PrimeCx)~ Odd(X))

3-✗ Prime(X) n – odd (X ) (True : 7=2

Negatingfx-J-xp.CH Pix = -✓x – )

– 3-✗ Unicorn (X)

Unicorn (x )

Quat-fyingour-ange.ofobjectsv-xc.IN

Fyc-IR: y¢☒ (some Real#saren’t rational )

Multi-argumentpredicatesv-x-vy-vzfmothercx.gl ) n Motherly, -2

Grandma (x ,z ) )

3-✗ try ✗ y = 0 (True , 21=0) read : There is an X which for

possible y values , ✗ – y proof by example .

7-✗thy ✗+y False .

We set the value of

and ask , is statement true ? No,

we can always find a

try3-✗ Xty= 0

Tru e . We imagine frying a ll possible values of y and , for each y

seeing if we can find an ✗

construction .

logicandproof.la

quantifiers

set time w e introduced

/for every X

there exists a n / there

We gave some examples with multiple

quantifiers a n d w e noted T.net

o rd e r the matter

of , quantifiers

✗ c- locks

lock has ita” opens .

Fyekeys : y

There is a key That opens

all locks ‘ ‘

2 n d is a very different statement!

Similarly : their ,

” Every Fyc- IR, theIR

” There is y3=✗

real#✗has a (TRUE )

a particular

fo r every

real # X . (FALSE)

IntroductionfoProofs_

mathematics a s

(truly infinite ) network of interconnected

ideas. A proof is like finding a path from o n e idea to the next .

5 -→ %”ifpthen ” “ifandonlyif”

often the path is not

obvious and w e

m ust find intermediate steps along the way

‘→ 4 → 2 → 2

s e a of built from

statements

a r e like islands of

truth amid bridges

contradiction .

other theorems

contradiction

definitions

, and A dragon

gp.g#Y+ucak

Statements

d. 1764 Moscow

A #statement is a sentence or mathematical

ex That is definitely true or false

U n k n o w n ‘s

c- 72 2+2=5 Goldbach’s

*£ 3 is even Prime113) then : even(x)

conjecture

N•+astatemÉwhich Prime(X) : The number ✗ is prime .

✗ > 5 : ✗ is greater than 5 . × ?

Example: V-a.b.snc- N, n>2 : a” +b” f-C”

” Fermat’s Last Theorem ”

: 1607- 1665

yygny.npyyyanay.am

A n d re w W ile s

1993 : ” Modular Forms , Elliptic curves, and Galois Representation,

÷÷÷÷::÷:÷:::

greater Than 2 is the sum of two prime numbers .

(nes ) → ftp.qc-P, n =p-18) WÉut

widely believed ftp.qc-P-n-P-qtobetrue#

T.hescarecrowbnewBro.int

he s u m of the square ro o ts of

any two sides of an isosceles triangle

is equal to the square root of

rq+ Ta = Fi (No)

V9+V4= Tg (No)

mathematical statement w o u l d

https://www.youtube.com/watch?v=IXpb-9fhoBM

3+97 41 -159

2 3 5 7 11 13 17 19

Consider ”

13 42008 4

17 4 3004 19 3

proofsw.thquantifiers-i-vx.tk/c-

F I E L specific

p.io?;i;ga,,a

X doesnt move

12–4 ⇒ ✗2=16 . : ✗2 iseven.

corre4Apmoad_ : suppose X is a n (arbitrarily chosen ) . a. n integer .

2k for some

KEZI (Defn of even)

:X ‘ is even (Defordeven)

Assume: show:

th Even Cx) → Even (X2) } Even Cx)

(Direct Proof)

Quantifiers : II-suffiT-H.ve

how to create it

Prove: F✗c-Even : ✗c-

Even Prime are , sets

EvenCx) n Prime(X)← Even

predicates ,

2 is divisible by only 1

i.2isanevenPrime •

arbitrarily

É Émostis

The worlds simplest proof ?

two arbitrary even

So ✗ + y = 2n -12m = 2 (ntm ) = 2k where K= n-1m

: ✗tyis even. Do

a) Prove sum of two odd integers is

Prove sum of even and odd is

hogicandproof-Direc-P.ro#

Proposition:p →

Thereforeq.

again at the truth table for p→g :

r¥¥F|P¥É} Implication i. automatically tru e if p is false

T f F } Implication is true if T T q is also true and

false if q is false

With direct proof w e apply modus

repeatedly :

what , would w e derive if p→q is false?

^ -(p→g)=pn-C-pug)

so deriving – of i.e , p n – q from assuming p

the implication i.e , – p → g) must

r-xample-V-xc.IOd d c x ) c → 0 d d ( ✗ 2 – 1 6 × – 1 8 )

✗ is odd iff ✗ 2-16×-18 is odd

we have to prove both : odd 4) → Odd (✗2+6×+8 ) Odd (X2-16×-18 ) → oddcx)

2A -11 (Defn of odd )

1)2 -1612A-11 ) -18

4 a ‘ +4 a -11 + 12 a -16 -18

= 4A? + 16a + 15 = 2(292+891-7)-1

where m=2a -18A-17

Therefore ✗2-16×+8 is odd (Defa of odd ) •

Contrapositive

How to prove: If✗2-16×-18 is oddthen✗isodd.

We could do another direct proof

but a contrapositive proof is simpler.

#P→Q Proposition :

w Outline :

Therefore – P •

Contrapositive of if ✗ 2+6×-18 is odd then ✗ odd .

If ✗ is not odd then ✗2+6×+8 is not odd .

if is II is

✗ e v e n , s o ✗ 2+6×-18

✗ = 2 a by definition

(2aYt6(2a)

4a2-112A -18

(292-1691-4)

= 2b where

b=2a46a -14

✗2-16×-18 is even

contrapositive example doesnt divide n ‘ -1 n

( with ) cases

n,n¥;uses?_

(works for neo too) 3 8

what is the contrapositive ?

If n is odd , 8 does divide n’

n=2a -11 Defn of

nh- I = (za-11)2-1 =

aora-11 iseven

1)a iseven: 4. 2m(2m-11)

= 8 m (2m-11

= 8k K= m(2m-11

ii)att is even : 4a(at 1) =4a(2m)

=8K(K=a.m)

proofbycontradictio.no

attire : Proposition.

proof : suppose

Thenefonec

P= -P→ (cn-

or (P→ (Cn- c))→ P=

it show you can

that – Pleads to a contradiction

This is a claim about a ll integers

c-Z, ifa’ iseven, a iseven

what is the

negation ?

Suppose: – the-2 (a”is even→ a is even)

– (a’ is even→a is even)

pefn’ ( – (- (a’ is even)v la iseven)p 21 77aiseven)nnlaiseven)

¥μafiFrFaeaeZ ( ‘

Thena = (2C1- = 4C +4C+I = 212C-12C)-11

Therefore a2 is odd

(K=2c2 -12C)

So a” isbothevenandodd (contradiction )

Prove : Tz irrational .

I 300’s BC (Euclid)

A real number ✗ is rational if

✗=&forsome pig

otherwise it is

irrational .

cancel out.

We further assume

¥ = 19-2–9-2

irrational

Proof ( By contradiction)

V2=g-⇒ 2=Pq÷or

So p’ is even , and so earlier proof. ) i.e , p=2k

is p. 82=2K’ ⇒qisalsoeven!

2g’ contradiction

both p and)q have a common factor(2)

: is irrational v2

aisratconalafdb.is/ratronaltheua.bisirational.Q

f- P V Q )

sappo-sei-N-l.ie

-(P→Q)or Pn-Q

Then there exists a , b

a is rational b is irrational and

Let a = myn mm c- 21

ab= My x.ge21

Thenab= 2¥=¥or b=¥1m

rational and irrational

( contradiction)

– (p → Q ) is

Therefore:P → Q

impossible

The-amaninfitenumber-fpr.ms

Note : Fundamental Theorem of Arithmetic :

integer greater than 1 is -either prime or can be made from

Unique product of prime factors .

1 0 4 ( composite

20= 2× 2✗5 2 52(composite)

13^4(compo

proof. Coy

) of contradiction

Then -there is last

✗ = (2) (3) (5) (7)

prime , call it p .

product of all primes

So each prime divides ✗

✗+ I = (2)(3)(5)- – – – (p) +1

✗+ I divisible by any prime ?

No ! For example , the next number that 2 divides into is ✗ -12 the next number that 3 divides is ✗-13

a re no prime factors

So X-11 is prime But ✗+1 >p .

the last Prime

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