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**preview**shows half of the first page. to view the full**3 pages of the document.**MATH 135 Winter 2009

Lecture IX Notes

If and only if

In mathematics, we often see statements of the form “Aif and only if B” (A⇔B). (See Assignment

1.) This means “(If Athen B) and (If Bthen A)”. The parentheses are here for mathematical

reasons, not English language ones!

Sometimes we say “The truth of Ais equivalent to the truth of B” or “Ais equivalent to B”, since

if A⇔Bhas been proven then if Ais TRUE, Bis TRUE, and if Ais FALSE, Bcannot be TRUE

(otherwise Awould 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=yif and only if x+y

2=√xy.

Proof

“⇒”

If x=y≥0, then x+y

2=2x

2=xand √xy =√x2=x(since x≥0) so x+y

2=√xy.

“⇐”

If x+y

2=√xy, then

x+y= 2√xy

(x+y)2= 4xy

x2+ 2xy +y2= 4xy

x2−2xy +y2= 0

(x−y)2= 0

x=y

Therefore, x=yif and only if x+y

2=√xy.

Example

In 4ABC,b=ccos(∠A) if and only if ∠C= 90◦.

Proof

“⇐”

If ∠C= 90◦, then cos(∠A) = b

c, so b=ccos(∠A).

“⇒”

Suppose b=ccos(∠A).

Drop a perpendicular from Bto Pon AC.

Then AP =AB cos(∠A) = ccos(∠A).

But AC =AP +P C and AC =b=ccos(∠A) = AP .

(Think about whether this makes sense if Pis to the right of C.)

Thus P C = 0, so Pand Ccoincide.

Therefore ∠BCA =∠BP A = 90◦.

B

C

A

a

c

b

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