Lecture: Prop. formulas, Truth tables,

Semantic Implications

theorem

Any Boolean function f : ht, Fb"→htt}

has a prop. formula rept. Yi ie .,

4) x,--o.

. . . * - un- flu

,- - un)

for any v , - . . une htt-

deffly

. ..vn#lVinuz)VV34--q,nxu)VXz

Let 52=4. - - xn} . A literal on r is

either x : or 7k.

Let uehtif} . Then × ;4=[Xiif he - T

TX , if U=F

is a literal.

this isjus atemp

.

"u ,I not.

Let ve hTiF3"

.

Consider N Xi = the[ =c

Let 's illustrate this.

Assume h=2 and u = (T , F)

x!' = x , X!- = 7×2 if -- X, 12×2Claim For any VELTRI?%/xi=a

. . . .x. men'T iff u=✓

Note that

far . -- un) -4¥; "" Haw. . . .#un

Xiu is a statement saying thot

Xi is equal to yDefinitionA formula 6 is in disjunctive normal form

if y -- Vhl Lise. if a disj of conj . of literals)A conj . of literals is called a teem

A formula 4 is in conj normal than it

4= Nl

And a disjunctions of literals is a clause.

ExetcisWe showed thot any Boolean functionhas a BNF show that

it has a CNF.

Hiryu (yet))mklxnD)n Glyn#→

→ tix#y) n Gyu 27)÷

Let's fix some f :#f}"→ 4TH

We showed that there an s sit.

>fly . . -un) lid)l*=u

. .. - xu --a

Note thot

( ( VA lij)) x.⇒y . . . xa=un==fveeiD)¥⇒. -*a

→ f Gq . .vn) = @V ' lis) Ix,=u . . . *uhflu ! ! -- un)