**1.** **Determine whether each of the following sets is well ordered. Either give a proof using the well-ordering property of the set of positive integers, or give an example of a subset of the set that has no smallest element.**

**a) the set of integers greater than 3**

**b) the set of even positive integers**

**c) the set of positive rational numbers**

**d) the set of positive rational numbers that can be written in the form a/2, where a is a positive integer**

**e) the set of nonnegative rational numbers**

**Solution:**

We begin by stating the **Well Ordering Principle**. It states that every non-empty set of positive integers has a smallest element.

**(a)** The set of Integers greater than 3 is well ordered as it is non-empty set of positive Integers and the smallest element in this set is 3.

**(b)** The set of even positive Integers is well ordered as it is non-empty set of positive Integers and the smallest element in this set is 2.

**(c)** The set of positive rational numbers is not well ordered. Let us take a subset of this set as

\left\{ 1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...\right\}
We can see that this non-empty positive subset has no least element. Instead it has a greatest element 1.

**(d)** The set of positive rational numbers of form \frac{a}{2} where a , is a positive integer is well Ordered.

The smallest positive Integer is 1 so the smallest positive rational number in this set is \frac{1}{2}

**(e)** The set of non-negative rational numbers is not well ordered. We take a subset of this set as the set of positive rational numbers which has a least element as can been seen from part (c).