How to write this in mathematical notation?

Question

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.

Answer 1

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 $y\in\mathbb{R}\setminus \mathbb{Q}$.

Answer 2

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".

Answer 3

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.