In
logic, there are four reformulations of any given
implication: the implication itself, the
converse, the inverse, and the
contrapositive.
For a given implication
a implies b, the inverse is formed by
not a implies not b.
Note that an implication and its contrapositive are
logically equivalent, while its converse and inverse are logically
equivalent. Note further that the converse and inverse are usually logically
unrelated to the implication and its contrapositive, except in the case of an
iff.
An example: "If it's raining then the streets are wet".
a is "it's raining" and
b is "the streets are wet". Here's a
truth table:
a | b | not a | not b | a --> b | not a --> not b | a <-> b
---------------------------------------------------------------------
T | T | F | F | T | T | T
T | F | F | T | F | T | F
F | T | T | F | T | F | F
F | F | T | T | T | T | T
The last column is "a if and only if b" which is true exactly when
a --> b and b --> a,
i.e. "a implies b and b implies a". For the iff statement, all four implication forms from above are
logically equivalent.