# Thread: A Quick Question on Valid Set Builder Notation

1. ## A Quick Question on Valid Set Builder Notation

Hello; today my Precalculus teacher gave this definition for the set of rational numbers:
$\displaystyle Q= \{ r|r= \frac{a}{b} ,a,b \in Z,b \not =0 \}$
Couldn't this be interpreted to mean that r equals both a/b and a, that b belongs to Z (the set of integers), and that b is not equal to zero?
Shouldn't it be:
$\displaystyle Q= \{ r|r= \frac{a}{b} , \{a,b \} \in Z,b \not =0 \}$
or
$\displaystyle Q= \{ r|r= \frac{a}{b} ;a,b \in Z,b \not =0 \}$?
Thanks,
yottaflop

2. ## Re: A Quick Question on Valid Set Builder Notation

Originally Posted by yottaflop
Hello; today my Precalculus teacher gave this definition for the set of real numbers:
$\displaystyle Q= \{ r|r= \frac{a}{b} ,a,b \in Z,b \not =0 \}$
Couldn't this be interpreted to mean that r equals both a/b and a, that b belongs to Z (the set of integers), and that b is not equal to zero?
Shouldn't it be:
$\displaystyle Q= \{ r|r= \frac{a}{b} , \{a,b \} \in Z,b \not =0 \}$
or
$\displaystyle Q= \{ r|r= \frac{a}{b} ;a,b \in Z,b \not =0 \}$?
Thanks,
yottaflop

$\displaystyle \mathbb{Q}$ is set of all rational numbers.

Here you can find some information:

Real number - Wikipedia, the free encyclopedia

Rational number - Wikipedia, the free encyclopedia

3. ## Re: A Quick Question on Valid Set Builder Notation

Hello, yottaflop!

A few spaces will provode clarity.

. . $\displaystyle Q \:=\: \{ r\,|\,r= \tfrac{a}{b},\;a,b \in Z,\;b \ne 0 \}$

I seriously doubt that a textbook would write it like this:

. . $\displaystyle Q\!=\!\{r|r\!=\!\tfrac{a}{b},\!a,\!b\!\in\!Z,\!b\! \ne\!0\}$

If it does, it may bring back public flogging.

4. ## Re: A Quick Question on Valid Set Builder Notation

Thank you both; Also sprach: that was a typo on my part, sorry for the confusion. I've edited my original post to fix it.

5. ## Re: A Quick Question on Valid Set Builder Notation

Originally Posted by Soroban
Hello, yottaflop!

A few spaces will provode clarity.

. . $\displaystyle Q \:=\: \{ r\,|\,r= \tfrac{a}{b},\;a,b \in Z,\;b \ne 0 \}$

I seriously doubt that a textbook would write it like this:

. . $\displaystyle Q\!=\!\{r|r\!=\!\tfrac{a}{b},\!a,\!b\!\in\!Z,\!b\! \ne\!0\}$

If it does, it may bring back public flogging.

I will direct you to this scrappy pack of textbook editors that have decided monkeys para-shooting out of planes and tightrope walking clowns will 'help' explain the math...

6. ## Re: A Quick Question on Valid Set Builder Notation

Originally Posted by yottaflop
Hello; today my Precalculus teacher gave this definition for the set of rational numbers:
$\displaystyle Q= \{ r|r= \frac{a}{b} ,a,b \in Z,b \not =0 \}$
=Couldn't this be interpreted to mean that r equals both a/b and a, that b belongs to Z (the set of integers), and that b is not equal to zero?
Shouldn't it be:
$\displaystyle Q= \{ r|r= \frac{a}{b} , \{a,b \} \in Z,b \not =0 \}$
or
$\displaystyle Q= \{ r|r= \frac{a}{b} ;a,b \in Z,b \not =0 \}$?
Thanks,
yottaflop
The second is better than what you initially give. The first of your suggestions is definitely wrong since it is requiring that the set, {a, b}, be an integer!