ICS 253: Discrete Structures I King Fahd University of Petroleum & Minerals Information & Computer Science Department ICS 253: Discrete Structures I The Foundations: Logic and Proofs
Section 1.7: Proof Methods and Strategy In Section 1.6 we briefly discussed the strategy behind constructing proofs. In this section, we will study some additional aspects of the art and science of proofs. Give advice on how to find a proof of a theorem. Describe some tricks of the trade, including how proofs can be found by working backward and by adapting existing proofs.
Exhaustive Proofs and Proof By Cases Exhaustive Proof: Some theorems can be proved by examining a relatively small number of examples. These proofs proceed by exhausting all possibilities. An exhaustive proof is a special type of proof by cases where each case involves checking a single example. Proof by Cases Prove that (p1 p2 … pn) q using the tautology [(p1 p2… pn)q] [(p1q)(p2q)…(pnq)]
Examples Prove that (n + 1)3 > 3n if n is a positive integer with n 4. Q4 pp. 102: Use a proof by cases to show that min( a, min( b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Show that there are no solutions in integers x and y of x2 + 3y2 = 8.
Without Loss of Generality When the phrase "without loss of generality" is used in a proof (abbreviated as WLOG), we assert that by proving one case of a theorem, no additional argument is required to prove other specified cases. That is, other cases follow by making straightforward changes to the argument, or by filling in some straightforward initial step. Use a proof by cases to show that |xy| = |x||y|, where x and y are real numbers.
Usually applied to theorems of the form xP(x). Existence Proofs Usually applied to theorems of the form xP(x). We usually prove them by Finding a particular element c such that P(c) holds, which is a form of constructive proof. Some other way, like proof by contradiction, which is a form of nonconstructive proof.
Examples Q6 pp 102: Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive? Q8 pp 102: Prove that either 2.10500 +15 or 2. 10500 +16 is not a perfect square. Is your proof constructive or nonconstructive?
Uniqueness Proofs To prove statements concerning the existence of a unique element, we show Existence: Uniqueness: Q17 pp 103: Show that if n is an odd integer, then there is a unique integer k such that n is the sum of k - 2 and k + 3.
Forward and Backward Reasoning Direct proofs are examples of forward reasoning. Unfortunately, forward reasoning is often difficult to use to prove more complicated results. Reasoning needed to reach the desired conclusion may be far from obvious. In such cases, it may be helpful to use backward reasoning. To reason backward to prove a statement q, we find a statement p that we can prove with the property that p q.
Example Given two distinct positive real numbers x and y, their arithmetic mean is (x + y) /2 and their geometric mean is . Prove that the arithmetic mean is always greater than the geometric mean. Proof:
Adapting Existing Proofs An excellent way to look for possible approaches that can be used to prove a statement is to take advantage of existing proofs. An existing proof can be adapted to prove a new result. Some of the ideas used in existing proofs may be helpful. Because existing proofs provide clues for new proofs, you should read and understand the proofs you encounter in your studies. Check the example on proving is irrational!
The 3x+1 Conjecture Let T be the transformation that sends an even integer x to x/2 and an odd integer x to 3x + 1. Conjecture: For all positive integers x, when transformation T is applied repeatedly, we will eventually reach the integer 1. For example, starting with x = 13, we find T(13) = 3 .13 + 1 = 40, T(40) = 40/2 = 20, T(20) = 20/2 = 10, T(10) = 10/2 = 5, T(5) = 3 · 5 + 1 = 16, T(16) = 8, T(8) = 4, T(4) = 2, and T(2) = 1. The 3x + 1 conjecture has been verified for all integers x up to 5.6 1013.