2. What is Mathematical Induction?
Mathematical induction is a method of proof that will
be used throughout your study. The basic theory is
this:
If we can prove that P(1) is true, and we can prove
that whenever P(𝑘) is true, when 𝑘 ∈ ℤ+ (a positive
integer), then P 𝑘 + 1 is true, then P(𝑛) is true for
all 𝑛 ∈ ℤ+
.
* Keep in mind that P 1 represents the least
element in the set. More on this…
3. The Well-Ordering Principle
Before we get too involved in mathematical induction, we must address the
issue with ℤ+
. ℤ+
refers to the set of numbers that include the positive
integers. You will sometimes see this as ℕ, which is the set of natural
numbers including the counting numbers; 1, 2, 3, …
So, let’s suppose that 𝑆 ∈ ℤ+, 𝑆 ≠ ∅, and that 𝑆 is bounded below.
The well-ordering principle says that 𝑆 has a least element.
These seems obvious. We have a subset of a discrete (countable) set, it isn’t
empty, and it is bounded below. Therefore, there must be a least element, or
an element in the set whose value is less than all other values.
Can we use mathematical induction with 𝑆 ∈ ℚ+
or 𝑆 ∈ ℝ+
?
4. Back to Mathematical Induction
Let P 𝑛 denote an open mathematical statement that involves one or
more occurrences of the variable 𝑛, which represents a positive integer:
If 𝑃(1) is true; and
If whenever 𝑃(𝑘) is true (for some particular, but arbitrarily chosen
𝑘 ∈ ℤ+), then 𝑃(𝑘 + 1) is true;
then P 𝑛 is true for all 𝑛 ∈ ℤ+.
Keep in mind that 𝑃(1) just represents the “least element”
