Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have the following claim:

“If $x$ and $y$ are real numbers and their product is irrational, then either $x$ or $y$ must be irrational.”

I'm supposed to write this in mathematical notation. It's my first year of university and I'm not so sure how to go about this, I never did any of this in high school.

Could anyone help me write this in mathematical notation?

I'm supposed to prove this later, but I have an idea how to do that, I just need help with this part. Thank you in advance.

share|improve this question
    
This whole question is troubling. Some math prof is asking you to take a clear and unambiguous English sentence and convert it into symbol-laden gobbledegook. For what purpose ??? –  bubba Sep 28 '13 at 2:21
    
You could even get rid of the $x$ and $y$, actually, and say "if the product of two real numbers is irrational, then at least one of them must be irrational". –  bubba Sep 28 '13 at 2:27
    
In your question, "either x or y" is misleading because it is possible for both x and y to be irrational and have their product be irrational too. –  Nayuki Minase Sep 28 '13 at 2:37
add comment

3 Answers

up vote 4 down vote accepted

I don't understand why this seems so difficult to the other people trying to give answers. I notice the discrete-math tag.

One want to write what you want is: $$ \forall x,y \in \mathbb{R} (xy\in \mathbb{R}\setminus\mathbb{Q} \to (x\in \mathbb{R}\setminus\mathbb{Q} \lor y \in \mathbb{R}\setminus \mathbb{Q})). $$

In "English" this is saying that: for all $x$ and $y$ real numbers, if the product of $x$ and $y$ is a real number, but not a rational number (i.e. $xy$ is irrational), then $x$ is a real number, but not a rational number (i.e. $x$ is irrational) or $y$ is a real number but not a rational number (i.e. $y$ is irrational).

You could use $\implies$ instead of $\to$. Also, some might prefer fewer parentheses.

Some also write $\mathbb{R} - \mathbb{Q}$ instead of $\mathbb{R}\setminus\mathbb{Q}$. This is simply the set of real numbers minus the set of rational numbers. In general $A\setminus B$ is the set of elements in $A$ that are not in $B$. So $\mathbb{R}\setminus \mathbb{Q}$ is the set of irrational numbers.

As mentioned in a comment by @HenningMakholm one might also prefer to write $x\notin \mathbb{Q}$ instead of $x\in\mathbb{R}\setminus \mathbb{Q}$. This, however is only good because we gave the domain as $\mathbb{R}$.

As also mentioned in other comments, while we read $\forall$ as for all, the symbol doesn't just replace the words. The symbol has a precise (mathematical) meaning. Likewise, $\lor$ mean or, but it is used between the two statements $x\in\mathbb{R}\setminus \mathbb{Q}$ and $y\in\mathbb{R}\setminus \mathbb{Q}$.

share|improve this answer
    
Could you explain this in English please? I get the gist of it, but I don't fully understand the "\" part..it would be very helpful if you explained that exactly as it is. @Thomas –  extremez Sep 28 '13 at 1:07
    
@extremez: Sure, let me try to update my answer to explain better. –  Thomas Sep 28 '13 at 2:27
    
Thanks, I now get it completely (: –  extremez Sep 28 '13 at 14:07
add comment

given the mistakes seen on other answers, let's try pure mathematical notation, although I always prefer a few words for both clarity and elegance...

$$\forall x,y\in\mathbb{R},xy\not\in\mathbb{Q}\Longrightarrow(x\not\in\mathbb{Q})\lor(y\not\in\mathbb{Q})$$

The parenthesis are not neccesary but I think tey give some more clarity.

share|improve this answer
add comment

I would write it as the contrapositive, because it's easier to make sense of: $$(x \in \mathbb{Q} \wedge y \in \mathbb{Q}) \implies xy \in \mathbb{Q}$$

But if you insist on that particular form: $$xy \notin \mathbb{Q} \implies (x \notin \mathbb{Q} \vee y \notin \mathbb{Q})$$

The $\in$ means "an element of", so $a \in S$ means "a is in the set S". The rationals are written as $\mathbb{Q}$, because Q is for quotient (R is taken for real numbers). The $\wedge$ and $\vee$ mean "and" and "or", respectively. Lastly, $\implies$ means "implies".

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.