## 2.3. The Principle of Induction

### Examples 2.3.2(a):

**S**is any set. Define*a b*if*a = b***S**is any set, and**P(S)**the power set of**S**. Define*A B*if*A B***S**is the set of real numbers between [0, 1]. Define*a b*if*a*is less than or equal to*b*(i.e. the 'usual' interpretation of the symbol )**S**is the set of real numbers between [0, 1]. Define*a b*if*a*is greater than or equal to*b*.

**1. S is any set. Define**

*a b*if*a = b*.
This is a trivial partial ordering. Since no element is related to an element
different from itself, this is not an ordered set. Without more information
about **S** we can not determine anything else.

This example shows that any set can be partially ordered.

**2. S is any set, and P(S) the power set of S. Define
A B if
A B**

Recall that if *A B* and
*B A* then *A = B*.
Therefore, this is indeed a partial ordering. Without more information about
the set **S** we can not determine anything else.

**3. S is the set of real numbers between [0, 1]. Define
a b if a is less than or
equal to b (i.e. the 'usual' interpretation of the symbol
)**

This is clearly an ordering, and the set [0, 1] with this ordering is usually represented as a subset of the number line. It is not a well-ordered set, because the subset (0,1] has no smallest element.

**4. S is the set of real numbers between [0, 1]. Define
a b if a is greater than or
equal to b.**

This is also an ordering. The set is not well-ordered, however, because the subset [0, 1) has no smallest element. Note that 1 is the smallest element in the set [0, 1), according to our convention. Here it is important to distinguish between the conventional meaning of the symbol and its meaning as we choose to define it for a particular situation.