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Home » Math Homework Help » Algebra Homework Help » Mathematical Induction Principal
Mathematical Induction Principal
The principle of finite induction (Axiom 5) has many important consequences. It gives us important principles for establishing the truth of certain propositions. The most important of them is the Principle of Mathematical Induction given by:

Theorem 1:
(First Principle of Induction): Let (T(n) | n N) be a set of statements, one for each natural number n. If the statement T(I) is true, and if the truth of T(k) implies that T(k + 1) is also true, then T(n) is true for all values of n N.

Proof: Let P be the set of all those natural numbers m for which T(m) is true, i.e.

P = (m | T(m) is true)

Since T(I) is true for 1 P and k P T(k) is true

T(k + 1) is true (k + 1) P.

Now it follows from the axiom of finite induction that P = N.

Hence T(n) is true for all n N.

Theorem 2: (Second principle of Induction): Let (T(n) | n N) be a set of statements, one for each natural number n. If

(i) T(I) is true and

(ii) For each natural number k > 1, the truth of T(m) for all m < k implies the truth of (T(k), then T(n) is true for all n N.

Proof: Let A be the set of all those natural numbers p for which T(p) is false, i.e.

Let A = {p | T(p) is false}

If A is non-empty, then A must have a least element, say, l (by the we ordering principle). This implies that if m < l then m ∉ A, i.e. T(m) is true. But the hypothesis (ii) implies that T(l) is true. Since T(l) is true, therefore, l ∉ A. But we standard with the assumption that l A. Hence we arrive at a contradiction, therefore, our assumption that A is non-empty is false. Hence T(n) is true for all n N.

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