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0.2 Logic and inference
9
A
=
⇒
B
means that whenever
A
is true,
B
must also be true, i.e., it CANNOT be
the case that
A
is true
B
is false: (
A
=
⇒
B
)
≡ ¬
(
A
and
¬
B
)
.
This means that the truth
table for =
⇒
can be found:
A
B
¬
B
A
and (
¬
B
)
¬
(
A
and
¬
B
)
T
T
F
F
T
T
F
T
T
F
F
T
F
F
T
F
F
T
F
T
=
⇒
A
B
A
=
⇒
B
T
T
T
T
F
F
F
T
T
F
F
T
A
B
¬
A
¬
B
A
=
⇒
B
¬
(
A
and
¬
B
)
¬
A
or
B
¬
B
=
⇒ ¬
A
B
=
⇒
A
¬
A
=
⇒ ¬
B
T
T
F
F
T
T
T
T
T
T
T
F
F
T
F
F
F
F
T
T
F
T
T
F
T
T
T
T
F
F
F
F
T
T
T
T
T
T
T
T
If
A
=
⇒
B
and
B
=
⇒
A
, then the statements are equivalent and we write “
A
if
and only if
B
” as
A
⇐⇒
B,A
≡
B,
or
A
iﬀ
B.
This is often used in
deﬁnitions
.
A
B
A
=
⇒
B
B
=
⇒
A
(
A
=
⇒
B
) and (
B
=
⇒
A
)
A
⇐⇒
B
T
T
T
T
T
T
T
F
F
T
F
F
F
T
T
F
F
F
F
F
T
T
T
T
If you know that
A
⇐⇒
B
, then you can replace
A
with
B
(or v.v.) wherever it
appears.
A
≡
B
is like “=” for logical statements.
One last rule (DeMorgan):
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 Fall '11
 Wong
 Logic

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