Algebra Know-It-ALL

148 Review Questions and Answers

2 = {0, 1} = {∅, {∅}}

3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}

4 = {0, 1, 2, 3} = {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}

Question 3-3

How is the set of natural numbers symbolized? What elements does it contain?

Answer 3-3

In this book, we symbolize the set of natural numbers as N, and define it as

N= {0, 1, 2, 3, 4, ...}

The three dots, called an ellipsis, tell us that the sequence goes on without end. The set N is

also known as the set of whole numbers. In some texts, N is defined without including 0:

N= {1, 2, 3, 4, 5, ...}

This is also called the set of counting numbers.

Question 3-4

According to the set-based definition of the natural numbers 0, 1, 2, 3, and so on, what num-

ber is represented by the entire set N?

Answer 3-4

Mathematicians call this an infinite ordinal or transfinite ordinal, and denote it using the

lowercase Greek letter omega (ω). We can imagine it as a form of “infinity.”

Question 3-5

How can we generate the set of even natural numbers (call it Neven) from the set N of natural

numbers? How can we generate the set of odd natural numbers (call it Nodd) from Neven?

Answer 3-5

We can generate Neven by taking each element of N and multiplying it by 2. We can generate

Nodd from Neven by taking every element of Neven and adding 1.

Question 3-6

In the set N, what is a prime number? What’s a composite number? Are there any natural

numbers that are neither prime nor composite? Are there any natural numbers that are both

prime and composite?

Answer 3-6

A prime number is a natural number larger than 1 (in other words, 2 or larger) that can only

be factored into a product of itself and 1. A composite number is a natural number that’s

a product of two or more primes. All the nonprime numbers larger than 1 are composite.

According to these definitions, the numbers 0 and 1 are neither prime nor composite. No

natural number can be both prime and composite.