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Mathematical Induction Principal

The principle of finite induction

Since

Now it follows from the axiom of finite induction that

Hence

Let

If

**(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: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**.**Services: -**Mathematical Induction Principal Homework | Mathematical Induction Principal Homework Help | Mathematical Induction Principal Homework Help Services | Live Mathematical Induction Principal Homework Help | Mathematical Induction Principal Homework Tutors | Online Mathematical Induction Principal Homework Help | Mathematical Induction Principal Tutors | Online Mathematical Induction Principal Tutors | Mathematical Induction Principal Homework Services | Mathematical Induction PrincipalSubmit Your Query ???

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