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6

**Not** every graph is connected : ¬(every graph is connected)

: **¬(∀x( Graph(x)⟹ Connected(x)))** *(option a)*

Now we can simplify above expression as below , **¬(∀x( ¬Graph(x)∨ Connected(x)))**

Now bringing negation inside,we get **∃x( Graph(x)∧¬ Connected(x))** (option b)

now apply double negation to the above statement as it does not change its meaning (law of double negation)..

**¬** **¬** (**∃x( Graph(x)∧¬ Connected(x)))**

now applying inner negation to all terms we get** :¬∀x(¬ Graph(x)∨ Connected(x)) **(option c)

Therefore option a,b,c are equivalent ..

answer:**D**

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0

20 votes

Just writing to simplify each options

A)¬∀x( Graph(x)⟹ Connected(x))

it is not the case that every graph then it will be connected,which implies that

some graph my be disconnected.

B)∃x( Graph(x)∧¬ Connected(x))

there exists some graph which are disconnected which implies

not every graph is connected.

C)¬∀x(¬ Graph(x)∨ Connected(x))

here no need to express it in english just solve first using

de morgn's law we will get it as option B)

D)∀x( Graph(x)⟹¬ Connected(x))

for all graph it is always disconnected...makes it FALSE

ANSWER D

A)¬∀x( Graph(x)⟹ Connected(x))

it is not the case that every graph then it will be connected,which implies that

some graph my be disconnected.

B)∃x( Graph(x)∧¬ Connected(x))

there exists some graph which are disconnected which implies

not every graph is connected.

C)¬∀x(¬ Graph(x)∨ Connected(x))

here no need to express it in english just solve first using

de morgn's law we will get it as option B)

D)∀x( Graph(x)⟹¬ Connected(x))

for all graph it is always disconnected...makes it FALSE

ANSWER D