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Understanding Strong Induction & Its Equivalence with Weak Induction & Well-Ordering - Pro, Study notes of Mathematics

University of Arizona (UA)Mathematics
Prof. David Savitt

Insights into the concept of strong induction and its equivalence with weak induction, the well-ordering principle, finite descent, and infinite descent. The authors use the metaphor of climbing a ladder to explain the difference between weak and strong induction. They also discuss the use of descent and infinite descent in proving statements. The document concludes by providing a solution to problem 12, which states that there are no positive integer solutions to x^4 + y^4 = z^2.

Typology: Study notes

Pre 2010

Uploaded on 08/26/2009

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Some notes on strong induction
Matt Ondrus and Ksenija Simic-Muller
March 7, 2007
(I found this at http://www-rcf.usc.edu/∼lototsky/PiMuEp/induct.pdf)
”I like to think of [weak] induction as sort of like climbing a ladder. The
base case is the bottom rung of the ladder and the induction step shows you
how to climb from one rung to the next. If you know how to get to the bottom
rung, and you know how to climb from one rung to the next, you can climb to
every rung above the bottom one! [...] The difference[between strong and weak
induction] is in what you need to assume in the induction step. For ordinary
induction in the ladder metaphor you simply go from the rung you are on up
to the next one. For strong induction, you need to know that all the rungs
below the rung you are on are solid in order to step up. As a practical matter,
both have the same logical strength when you apply them since as you climb up
the ladder from the bottom rung, you sweep through all the intermediate rungs
anyway.”
There are other useful statements related to induction:
•The well-ordering principle; it states that every nonempty set of natural
numbers has a least element.
•The principle of descent; you will see it stated in different forms, but, in
simplest terms, it states that if Pis a property of natural numbers, and
if there is a counterexample to P, then there is the least counterexample.
We usually use descent to prove the contrapositive of a statement: suppose
you want to prove that Pholds for all natural numbers; assume there is
a number nfor which it fails; by descent you can assume that it is the
smallest such; if we are able to derive from this that there is a smaller
value for which Pfails, then we get a contradiction, so Pis in fact always
true.
•The principle of infinite descent states that: If Pis a property of natural
numbers, and if, with the assumption that there exists an nfor which
P(n) is true you could construct a sequence n > n1> n2> . . . for which
Pholds, then Pin fact fails for all n.
1
pf2

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Some notes on strong induction

Matt Ondrus and Ksenija Simic-Muller

March 7, 2007

(I found this at http://www-rcf.usc.edu/∼ lototsky/PiMuEp/induct.pdf) ”I like to think of [weak] induction as sort of like climbing a ladder. The base case is the bottom rung of the ladder and the induction step shows you how to climb from one rung to the next. If you know how to get to the bottom rung, and you know how to climb from one rung to the next, you can climb to every rung above the bottom one! [...] The difference[between strong and weak induction] is in what you need to assume in the induction step. For ordinary induction in the ladder metaphor you simply go from the rung you are on up to the next one. For strong induction, you need to know that all the rungs below the rung you are on are solid in order to step up. As a practical matter, both have the same logical strength when you apply them since as you climb up the ladder from the bottom rung, you sweep through all the intermediate rungs anyway.” There are other useful statements related to induction:

  • The well-ordering principle; it states that every nonempty set of natural numbers has a least element.
  • The principle of descent; you will see it stated in different forms, but, in simplest terms, it states that if P is a property of natural numbers, and if there is a counterexample to P , then there is the least counterexample. We usually use descent to prove the contrapositive of a statement: suppose you want to prove that P holds for all natural numbers; assume there is a number n for which it fails; by descent you can assume that it is the smallest such; if we are able to derive from this that there is a smaller value for which P fails, then we get a contradiction, so P is in fact always true.
  • The principle of infinite descent states that: If P is a property of natural numbers, and if, with the assumption that there exists an n for which P (n) is true you could construct a sequence n > n 1 > n 2 >... for which P holds, then P in fact fails for all n.

The following are equivalent:

  • Weak induction
  • Strong induction
  • Well-ordering principle
  • Finite descent
  • Infinite descent.

By equivalent, we mean that they all have the same logical strength, i.e. can prove the same statements (though some logicians argue that infinite descent is different from the other four). Both descent and infinite descent can often be used when it is not clear how to use weak or strong induction, as in the next example. Solution of Problem 12: There are no positive integer solutions of x^4 + y^4 = z^2. Use the fact that every positive integer solution of a^2 + b^2 = c^2 where a, b and c are relatively prime can be expressed in terms of two relatively prime numbers m and n where a = m^2 − n^2 , b = 2mn, and c = m^2 + n^2. (Notice that we can assume that a, b and c are relatively prime, since if they are not, we can divide the whole expression by their GCF; also notice that exactly one of a, b is even, so assume that this is b.) So suppose you have a solution to x^4 + y^4 = z^2. Applying the above fact to the pythagorean triple x^2 , y^2 , z gives x^2 = p^2 − q^2 , y^2 = 2pq, and z = p^2 + q^2. Since 2pq is a square, and p and q have no common factors, one of p and q must be 2 times a square and the other is an odd square (convince yourself this is true). And since x, p, q themselves form a (relatively prime) pythagorean triple (since x^2 +q^2 = p^2 ), we can see that there are r and s for which x = r^2 −s^2 , q = 2rs, and p = r^2 +s^2. This means q is the one that is 2 times a square (q = 2u^2 ) and p is an odd square (p = v^2 ). Now q = 2u^2 = 2rs, which means r and s are themselves perfect squares (because they are relatively prime). Yet r^2 + s^2 = p = v^2 which means v^2 is expressible as the sum of fourth powers (since r and s are perfect squares). Yet if we look at the construction of v, we see v^2 < z^2 , which creates our infinite descent (since you can repeat the process again), a contradiction. You can also think of this problem in terms of finite descent, if we assume that (x, y, z) is the smallest triple for which the claim holds. Logicians would point out here that we were being sloppy, because we are creating an infinite descending sequence of triples, and not natural numbers, as is required, but this can easily be remedied, by using some form of coding that would assign to each triple a single number and would preserve ordering. Such codings do exist (”G¨odel numbering”).