the statements are:
I) If the cat does not have claws, then the cat cannot scratch the furniture.
II) If the cat can scratch the furniture, then the cat has claws.
III) If the cat has claws, then the cat can scratch the furniture.
you can solve this using truth tables.
you would do so as follows:
p = cat has claws
q = cat scratches furniture
~p->~q = I
q->p = II
p->q = III
truth table:
p q ~p ~q ~p->~q q->p p->q
T T F F T T T
T F F T T T F
F T T F F F T
F F T T T T T
from the truth tables, it appears that ~p->~q is equivalent to q->p because their truth tables are the same.
that means that I is equivalent to II
III is not equivalent to either one of them because its truth table is different.
if we examine the statement, we will see why this is so.
the statement are, once again:
I) If the cat does not have claws, then the cat cannot scratch the furniture.
II) If the cat can scratch the furniture, then the cat has claws.
III) If the cat has claws, then the cat can scratch the furniture.
statement II looks like it is the contrapositive of statement I.
statement I says:
if no claws, then no scratch furniture.
statement II says:
if scratch furniture, then claws.
if the statement is:
if no claws, then no scratch furniture
then the contrapositive to that statement is:
if scratch furniture, then claws.
if the statement is:
if scratch furniture, then claws
then the contrapositive to that statement is:
if no claws, then no scratch furniture.
the statement and its contrapositive are equivalent which means if the statement is true then the contrapositive is also true and vice versa.
this is the case between statements I and II.
statement III says:
if claws, then scratch furniture.
the contrapositive to this would be.
if no scratch furniture, then no claws.
all 3 statements are:
I = if no claws, then no scratch furniture
II = if scratch furniture, then claws.
III = if claws, then scratch furniture.
the contrapositive to statement III is:
if no scratch furniture, then no claws
neither I or II are the contrapositive to statement III, therefore the only equivalent statement are I and II.
It's a little convoluted, but that's the way it works out.
Use of the truth tables can make sense of the insensible, assuming you know how to use them.
p->q is only false if p is true and q is false.
q->p is only false if q is true and p is false.
~p->~q is only false if ~p is true and ~q is false.
~q->~p is only false if ~q is true and ~p is false.
that's the truth table for the implied statement.l