## 8.3.Mathematical Induction

Definition 7.2 (Induction)

Assume that the universe of discourse is given by the set of the natural numbers, and let P(n) be any property of natural numbers. In this case we can use the following rule of inference.
```P(0)
for all n ( P(n) ) -> P( s(n) )
------------------------------------
for all n P(n)
```

P(0) is called the inductive base or the basis of induction.

The expression for all n( P(n) ) P( s(n) ) is called the inductive step.

Hence, the inductive base and the inductive step together imply that P(n) is true for all n.

This leads us to the following method which is called strong induction:

```Inductive base:         Establish P(0)
Inductive hypothesis:   Assume P(0), P(1), P(2), ... P(n)
Inductive step:         prove P(n+1)
Conclusion:             The basis of induction and the inductive step
allow one to conclude that P(n) is true for all n.
```

Examples:

• Prove that for n Nat.
• Prove that for a 1 and n Nat.
• Prove that the sum of the first n odd integers is for n Nat
• Prove that for n 3 and n Nat
• Prove that for n 4 and n Nat
• Prove that for n 4 for n Nat
• Prove that for n Nat.
• Prove that (n+2)! is even for n Nat.
• Prove that 0 is left zero for multiplication. In other words, prove that 0 × n = 0, for n Nat
• Prove that for n 3 and n Nat.
• Prove that for n 4 and n Nat.
• Prove that for a 1 and n Nat.
• Prove that for n Nat.