SOLUTION: Write the converse, inverse, and contrapositive of the following conditional statement If a dog is barking, then it will not bite

Write that as BARKING -> ~BITE

Then use the rules:

1. To form the converse conditional of a given conditional,
exchange what's on the left of the -> with what's on the right.
(But do not NEGATE either one.)

So the converse conditional of the given conditional
BARKING -> ~BITE

is

~BITE -> BARKING, or

If a dog will not bite, then it will be barking.

Note: I changed "is barking" to "will be barking" to make
the sentence clearer without changing the meaning. You can
change the tense of the verbs without changing the meaning.

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2. To form the inverse conditional of a given conditional,
NEGATE both what's on the left of the -> as well as what's 
on the right. (But do not exchange them). To NEGATE, if 
there is a ~ before it, take it off, and if there is no ~ 
before it, put a ~ before it.

So the inverse conditional of the given conditional
BARK -> ~BITE

is

~BARK -> BITE, or

If a dog is not barking, then it will bite.

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3. To form the contrapositive conditional of a given conditional,
EXCHANGE what's on the left of the -> with what's on the right AND
NEGATE them also.

So the contrapositive conditional of the given conditional
BARK -> ~BITE

is

BITE -> ~BARK, or

If a dog will bite, then it will not be barking.

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Notice also these facts that will help you understand: 

1. the original statement and the contrapositive 
   are equivalent statements.
2. the converse and the inverse are equivalent
   statements.
3. the original and its equivalent contrapositive ARE NOT
   equivalent to the equivalent inverse and converse.
4. the inverse is the contrapositive of the converse.
5. the converse is the contrapositive of the inverse.

Edwin