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→

statement

:

to

be

assigned

HE

values

: denote with small letters

eg

.

:

a

number between

and

1 × 1

implication

:

p→g

is

true

Lp

is

false

pandas

are both

true

double

implication

:

page

is

true

{

both the

ii.

e.

same

truth

value

both

false

1

Ex.

Suppose

• a

→

b

c)

is

F. determine

all

possible

auth

value

of

lav

b)

A

1- b

V

• a → Cba

c)

is

f

'

n

na

is

Tiff

e s

c)

is

f

i. a

is

F

{

do

is

Tiff

ne

ist

c

cis

b is

7- iffnc

is

Tlc

is

f)

on

.

of c

nb

a

c

avb

nb

Vnc sensu

1 I

0

0

0 O o

l

l

to

The

stint

is

always

false

contradiction

→

given

implication

p

→

q

:

converse

:

g→p

contrapositive

:

z→

p

cerey

thing

switched

implication

and

its

contrapositive

is

d. e

,

so

,

to show

a

start

is

true

or

false

,

you

can show its

contrapositive

is true

or

false

inverse

:

p

→

an

contrapositive

of

converse

,

not

common

)

Start

p

is

sufficient for

btmt

2

:

if

the

truth

of

p

guarantees

truth

if I :p

→

g

Sans

p

is

necessary

for

start

g.

if

p

must be

true

in

order

foczto

be

true

:

I

→

p

Necessary

is

converse

of

sufficient

Eoe

. ofonslate

into

symbols

:

"

"

In order to do a

Kiat

hon

,

you

must

be

able to swim

:

t

:

you

do a

triathlon

s

:

you

can

swim

t

→

s

I

do triathlon must can

swim

,

but

can

swim not

mean

can

do

a triathlon

)

prove

:

e-

passive

pnglrc page

c- pncsgirq

)

repay

factors

e)

Phil V

past

a

prepng

) i distribution )

pvp

in

c-

pray

E)

a

c-

prop

spray

% turn

"

a

"

to

"

v

"

and

"

"

or

to turn

"

v

"

to

"

a

"

and

"

a

"

,

ase

double

negation

and

then

Dreltogan

's Law

→

Logical Implication

:

p

→

I

,

i.

e.

,

p

→

gis

tautology

,

written

as

p g

e.

g.

:

to

prove

png p

:

P

P^q

PIPP

0

0

0 I

0 I

'

l

0

\

.

. 1

,

so

page

p

'

9 :

1

or use laws

:

pay

→

p

• c

page up

pushup

zvcnpvp

'

EV

I

I

i.

page p

→

Polynomial

kills

log

exponential

kills

polynomial

Ex

.h¥→¥

for

any

171

,

er

7 ¥

it

at

D

Ex :

Sn =

It

it

to

"

t

n

,

what

is

Sn

Sn

: It

'

t

n

y

= at

..

.tl/2Sn=K-l)n--sn=Cntyn-

Ex :

Un

• • l

'

ti t

.

'

n

, prove

Kazi

,

Un

(

^M+#j

base

:

n

• • l i

Un

• • l

=

5

= 1

✓

suppose

the

= ¥ 15

it

Kun

We

Want

A

K+

,

=

Kk+blkI4_j

?

Ukti

=

Uk

yet

=

IÉKt

=

(

Kt

lie

• (

KY 4kt 41

=

1kt1)

~ • (

Ktvt

i. In =

fÉJ

fer

all a

H

"

Sn

=

Is

a

r

,

claim

Sn

=

a

lMf¥)

:

Base

:

a-

0

Sn

=

a

91-

:

Sk

:

rE

a

fir

some

Kuo

95

:

Sui

=

r*r

a?

9kt

I

=

skta.pk

=

a

(

r

"

,

'

r

'

" '

)

ft

"

r

• I -

• I

= a

r

I

p

141

.

r

I

=-

h

r

I

pk

l

= a

r

I

i. Sn

a V"r¥

)

for

all n

so

the

analogy of

this

is

feed

✗

=

et

• c

→

Floors and

ceilings

:

I

LX)

=

c-

Z

I 2-

e-

✗

a

(ex

. C- 4.

=

• I

<

1 × 1

E

X

E

1Mt

XH

I

a)

= t

if

I

z

• I

<

✗

EZ

le

✗.

f-

4-

=

floor

:

largest

c-

✗ i

if

not

integer

:

the

nearest

to the

left

of

✗

ceiling

: smallest

i

if

✗

not

integer

:

the

neatest to

the

right

of

The

floor

/

ceiling of

an

integer

is

itself

representing

a

number in fade

f

:

every

digits

:

i o ,

1

,

..

.

,

f-

1

's

change

to

base to

: Onda - i

do

the

= If

°

,

f

'

... + dnfn

" D:

to

,

,

,

,

f

:

A

→

A

=

I

"

co

)

=

T

= "

calculate

f-

I

f-

'

: 0

fly

y

=

I

-1131=

f-

'

= 2

-1141=

f-

'

=

:

R

→

R

J

LX )

=

f

is

injective

,

subjective

fond

f-

1

injective

:

fan

fix

.

×

?

-15=+1+

✗

i. =

i

Xv

subjective

:

for

any

y

c-

R

.

y

• • ✗

45

1=4-

×

:

toys

c- R

y

:

=

'

F-

i.

g-

¥ 5

✗ c-

A)

N

=

1

,

2

,

---

)

"

"

"

÷÷¥÷÷÷

.

"

I

=

1 ¥ ,

PEZ

,

LE

2.2=

tzxt

:

!

!!

→

0,

a

bijection

'

' '

I 3

✗

: N

→ Z

Z - I

o

l

L

il

a

2- ✗ Z

countable class

: : :

:

"

spider

1

g-

n

o

,

to

3

'

iii. ¥

;

:

method

"

☐

Same

cardinality if

affection

f

: ✗

→

y

is

inject

ice

"

1 × 1

"

Éiyi

"

:

→

y

is

surjective

"

1 × 1

"

"

lip

"

→

Strategies

for proving

countability

:

fist

surjection

f.

N

✗ C-

Some countable

set

Union

of

countable

sets

are

also countable