MATH 135
Winter 2009
Lecture IX Notes
If and only if
In mathematics, we often see statements of the form “
A
if and only if
B
” (
A
⇔
B
). (See Assignment
1.)
This means “(If
A
then
B
) and (If
B
then
A
)”.
The parentheses are here for mathematical
reasons, not English language ones!
Sometimes we say “The truth of
A
is equivalent to the truth of
B
” or “
A
is equivalent to
B
”, since
if
A
⇔
B
has been proven then if
A
is TRUE,
B
is TRUE, and if
A
is FALSE,
B
cannot be TRUE
(otherwise
A
would be). Can you see why?
To prove these statements, we have two directions to prove, since there are two “If...then...” state
ments that must be proven to be TRUE.
Example
Suppose
x, y
≥
0. Then
x
=
y
if and only if
x
+
y
2
=
√
xy
.
Proof
“
⇒
”
If
x
=
y
≥
0, then
x
+
y
2
=
2
x
2
=
x
and
√
xy
=
√
x
2
=
x
(since
x
≥
0) so
x
+
y
2
=
√
xy
.
“
⇐
”
If
x
+
y
2
=
√
xy
, then
x
+
y
=
2
√
xy
(
x
+
y
)
2
=
4
xy
x
2
+ 2
xy
+
y
2
=
4
xy
x
2

2
xy
+
y
2
=
0
(
x

y
)
2
=
0
x
=
y
Therefore,
x
=
y
if and only if
x
+
y
2
=
√
xy
.
Example
In
ABC
,
b
=
c
cos(
∠
A
) if and only if
∠
C
= 90
◦
.
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 Winter '07
 peter
 Math, Logic, Mathematical logic, Proof by contradiction, Reductio ad absurdum, Contrapositive Recall

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