Pure 1.8 - Proof by induction

Question 1

What is proof by induction used for?

Answer 1

Proof by induction is used to prove that a general statement is true for all positive integers.

Question 2

Describe the general process for a proof by induction.

Answer 2

Proof by mathematical induction usually consists of the following four steps:
Step 1 - Basis: Prove the general statement is true for 𝑛=1.
Step 2 - Assumption: Assume the general statement is true for 𝑛=𝑘.
Step 3 -Inductive: Show that the general statement is then true for 𝑛=𝑘+1.
Step 4 - Conclusion: The general statement is then true for all positive integers, 𝑛.
It is possible to use this process to show the statement is true though showing that a given expression is divisible by a certain integer.

Question 3

What is meant by ℤ⁺?

Answer 3

ℤ⁺ is the set of positive integers, 1,2,3,… It is equivalent to ℕ, the set of natural numbers.