Chapter 2.1, Problem 40WE

a.

To determine

To write: whether set of positive integers are closed under $÷$

Answer to Problem 40WE

No, positive integers are not closed under $÷$

Explanation of Solution

Given information: A set of numbers is said to be closed under an operation if the result of combining any two numbers in the set results in a number that is also in the set

Take $a=2,b=3$

Here, $2,3$ are positive integers.

Now find $2÷3$

  $2÷3=\frac{2}{3}$

But $\frac{2}{3}$ is not an integer.

So, positive integers are not closed under $÷$

b.

To determine

To write: whether set of odd integers are closed under $\times$

Answer to Problem 40WE

Yes, set of odd integers are closed under $\times$

Explanation of Solution

Every odd integer is of the form $2p+1$ where $p$ is any integer

Let $a,b$ be any two odd integers.

$a=2m+1,b=2n+1$ where $m,n$ are integers.

Now find $ab$

  $\begin{array}{c}ab=\left(2m+1\right)\left(2n+1\right)\\ =2m\left(2n+1\right)+1\left(2n+1\right)\\ =4mn+2m+2n+1\\ =2\left(2mn+m+n\right)+1\end{array}$

Let $k=2mn+m+n$

So,

  $ab=2k+1$

So, $ab$ is also an odd integer.

c.

To determine

To write: whether set of odd integers are closed under $+$

Answer to Problem 40WE

No, set of odd integers are not closed under $+$

Explanation of Solution

Take $a=3,b=5$

Here, $3,5$ are odd integers.

Now find $3+5$

  $3+5=8$

But $8$ is not an odd integer.

( $8$ is an even integer)

So, odd integers are not closed under $+$

d.

To determine

To write: whether set of integers ending in 4 or 6 are closed under $\times$

Answer to Problem 40WE

Yes, set of integers ending in 4 or 6 are closed under $\times$

Explanation of Solution

Let $a,b$ be any two integers ending in 4 or 6.

Then $ab$ is also an integer ending in 4 or 6

For example:

Take $a=14,b=24$

  $ab=(14)(24)=336$

So, $ab$ ends in 6.

Therefore, set of integers ending in 4 or 6 are closed under $\times$