Quick Question on Sets: Sketching H = {(x,y) : xy = 0} - Explained

[Benny]

Hi I'm just wondering how I would sketch the following, it's part of a subspace question.

H = {(x,y) : xy = 0}

I don't thinking dividing my x will do that much. If x is not equal to zero then dividing by x gives me y = 0. But what would happen for points (x,y) where x = 0?

It's one of the questions in my question booklet, not an assignment question. I wouldn't post an assignment question on the internet. Any help would be appreciated.

 

[TD]

Well, xy = 0 \Leftrightarrow x = 0\,\, \vee \,\,y = 0

 

[Benny]

I am not familiar with the notation that you have used. Does the v mean "or?" Thanks for the help.

 

[TD]

Yes, indeed.
A product of two factors is 0 iff at least one of them is 0.

If x isn't zero, you can divide by x and find that y has to be zero.
If y isn't zero, you can divide by y and find that x has to be zero.
In the last case both are zero, and then you have the element (0,0)

 

[Benny]

Thanks again, what you've said makes sense.