In my Discrete Math lecture, we were told that we should omit any number that repeats in the set of rational numbers $\mathbb{Q}$. In other words, we should only keep $\frac{1}{1}$ and omit $\frac{2}{2}, \frac{3}{3}$, etc. But I don't see why this would be the case, since the set is defined as $\mathbb{Q} = \{\frac{a}{b}|a,b\in\mathbb{N}, b\ne 0\} $ Does the set $\mathbb{Q}$ include these repeating values?
Asked
Active
Viewed 167 times
2
-
@JoséCarlosSantos link is sufficient. If it also helps, you can also attach the addendum to your set that $\gcd(a,b) = 1$ if that makes you feel better. – Gregory Oct 13 '22 at 17:58
-
Not exactly. Sure, the elements are the same value wise, but that's only if you choose to simplify each element. If you don't, then would you technically have distinct elements $\frac{1}{1}, \frac{2}{2}, \frac{3}{3}$? – chefravioli Oct 13 '22 at 17:58
-
"the elements are the same value wise" Yes. They are the same... so even if we wrote them multiple times that would just be redundancy. The set ${1,1,1,1,1,2}$ is a two-element set. It is equal to the set ${1,2}$. Whether we were redundant or not, we only ever consider something to "be in" the set or "not be in" the set. There are no alternatives and there is not a concept when it comes to sets about "being in a set multiple times" – JMoravitz Oct 13 '22 at 18:02
-
3The value of an element does not depend of the way you write it. $\frac11=\frac22.$ A rational number $\frac ab$ is not an ordered pair $(a,b)$. – Anne Bauval Oct 13 '22 at 18:03
-
After looking around with some different wording, this post answers my question: https://math.stackexchange.com/questions/3346612/are-equivalent-fractions-in-a-set-considered-distinct-elements. @AnneBauval answered the question in the same way. – chefravioli Oct 13 '22 at 18:05
-
1Now... if you wanted to be more formal... you should be defining $\Bbb Q$ as the quotient ring $(\Bbb Z\times (\Bbb N\setminus{0}))/\simeq$ where $\simeq$ is the equivalence relation $(a,b)\simeq (c,d)\iff ad = bc$. We choose to write $\frac{a}{b}$ rather than $(a,b)$ for aesthetic purposes. Since $\Bbb Q$ is a quotient, the elements are technically equivalence classes. Your saying $\frac{1}{1}$ is an element of $\Bbb Q$ is more accurately saying that $[\frac{1}{1}]$ is an element of $\Bbb Q$, and indeed $[\frac{1}{1}]$ is equal as a set to $[\frac{2}{2}]$ – JMoravitz Oct 13 '22 at 18:06
-
Keep in mind that, for example, ${\frac{1}{2},,\frac{2}{4},,\frac{3}{6}}={\frac{1}{2}}$. It does not matter how many times you list an element in a set, it can only be in the set once. Either it is in the set or it is not in the set. There is no other option. – John Wayland Bales Oct 14 '22 at 06:45
1 Answers
1
That is a nice question. You see, when we build the set $\mathbb{Q}$ we do not only define the elements in form, there is an additional definition of "equality". Let $x=\frac{a}{b}$ and $y=\frac{c}{d}$. We say that $$x=y\Leftrightarrow ad=bc$$
So in fact, the set of rational numbers is the set $$\left\{\frac{a}{b}\vert a\in \mathbb{Z}\wedge b\in \mathbb{Z}^\ast\right\}$$ after we identify i.e. "glue" the equal numbers together.
