Algebra Know-It-ALL

Question 2-10

How can we describe the relationship between the following sets? How would we write it in
symbolic form?

A= {4, 5, 6, 7, 8, 9, 10, ...}

B= {7, 8, 9, 10, 11, 12, ...}

Answer 2-10

In this case, B is a proper subset of A. That’s because every element in B is also in A, but there
are some elements in A that are not in B. We write this fact as B⊂A.

Chapter 3

Question 3-1

The natural numbers are sometimes defined in terms of sets. How can we do this?

Answer 3-1

We can define the number 0 as the set containing nothing. That’s the null set:

0 = { } =∅

Once we’ve defined the number 0, then we can define the number 1 as the set containing the
number 0, like this:

1 = {{ }} = {∅}= {0}

After that, we can build the rest of the natural numbers upon each other:

2 = {0, 1}

3 = {0, 1, 2}

4 = {0, 1, 2, 3}

↓

n+ 1 = {0, 1, 2, ..., n}

↓

and so on, forever

Question 3-2

How can we write the natural numbers 0 through 4 purely in terms of set braces and the null
set symbol?

Answer 3-2

We start with 0, which is equal to ∅ by definition. The numbers are built upon each other as
sets within sets, like this:

0 =∅

1 = {0} = {∅}

Part One 147