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Proof Techniques: Addendum
When to use a Direct Proof and when to use Indirect Proof
Suppose you are to prove the following.
If n is even then n2 is even.
(1)
Here clearly a direct proof is preferable since we can start with an even number n=2i, for some
integer i. Then n2 = 4i2=2 (2i2). Therefore, n2 is even.
Now consider this problem.
If n2 is even then n is even.
(2)
Here, if we start with n2 = 2i and then take the square root, we get n = sqrt(2)* sqrt (i), which
we cannot claim as being even. Thus we are stuck.
Next, we try an indirect proof. An indirect proof for p  q is to show not q  not p. Thus we
have to show that if n is not even then n2 is not even.
If n is not even then n = 2i+1 and thus n2 = (2i+1)*(2i+1) = 2(2i2 + 2i) + 1 (this is the form for an
odd number)  n2 is not even.
On a separate note the statements (1) and (2) above can be combined by saying,
n is even if and only if n2 is even.
Or, equivalently, it is a statement of the form P if and only if Q ( P  Q ).
This statement says that P and Q are logically equivalent.
a) If P is true then Q is true (P  Q)
and
b) If P is false then Q is false (or equivalently, Q  P. Why? Answer: Think of the indirect
proof for b).
Important Note: P  Q is sometimes written as Q if P or P is a sufficient condition for Q.
Also we can write P  Q as P only if Q or Q is a necessary condition for P.
It Seems Too Easy
If you find a simple proof, and you are convinced of its correctness, then don't be shy
about. Many times proofs are simple and short.
In the theorem below, a perfect square is meant to be an integer in the form x2 where x
itself is an integer and an odd integer is any integer in the form 2x+1 where x is an
integer.
Theorem. Every odd integer is the difference of two perfect squares.
Proof. Suppose 2x+1 is an odd integer, then
2x+1 = (x+1)2 - x2.
Where's the proof? It's there. Just expand the RHS and you see that it is equal to the LHS.
The above proof is an example of a constructive proof, because the proof gives
(constructs) the two perfect-square numbers whose difference is equal to the odd number.
For example, 17 (17=2x+1  x= 8); thus 17 = 92 - 82.
Another Simple Example
Recall that a natural number is called composite if it is the product of other natural
numbers all greater than 1. For example, the number 39481461 is composite since it is
the product of 15489 and 2549.
Theorem. The number 100...01 (with 3n-1 zeros where n is an integer larger then 0) is
composite.
Proof. We can rewrite our number as 100...01 = 103n + 1 where n is an integer larger than
0. Now use the identity a3 + b3 = (a+b)(a2 - a b + b2) with a = 10n and b = 1, to get
(10n)3 + 1 = (10n + 1)(102n - 10n + 1).
We will be done once we have shown that both factors (10n + 1) and (102n - 10n + 1) are
greater than 1. In the first case, this is clear since 10n > 0 when n > 0. In the second case,
102n - 10n = 10n (10n - 1) > 0, when n > 0. This completes the proof.