QUESTION

$2$

$[$

$3$

$0$

$]$

Use the logical equivaleceses above to show that

$-$

$($

p

V

$-$

$($

$\backslash$rho

q

$)$

$)$

$$is a contradiction.

$[$

$3$

$]$

Use a truth table to verify the first De Morgan law not

$($

p

q

$)$

~~notpvvnotq.

Show that not

$($

p

v

v

$($

n

o

t

p

q

$)$

$)$

$$and

n

o

t

p

notq are logically equivalent by develoging a series of logical equivalences.

Show that pvva

$->$

r

$=$

$($

p

$->$

r

$)$

$($

a

$->$

r

$)$

$$using a series of logical equivalences.

$5$

$$Assume tat

A

$,$

B

$,$

$$and C are atemic sentences and are not logically independent. Assume firther that

A

$$is necessurily false

$($

$$cog$,$

A

$$is the prodicate Smaller

$($

s

$,$

a

$)$

$)$

$.$

$$Determine whether the following sentences are satisfiable or logically true:

S

$.$

$1$

$$Avnot

$($

B

v

v

$-$

$($

C

A

$)$

$)$

$[$

$4$

$]$

Prove:

p

$->$

q

$<=$

$>$

$-$

q

$->$

$-$

p

$$using a series of logical equivalences.

$($

$5$

$)$

$($

$8$

$)$

$$Use Doublo Negation rule, DeMargen's rules and other Derivation Rules to prove that the following sentences are logically equivalees.