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u and v can have components (they are vectors in 2D) that are both positive real numbers or 0, or u and v can have components that are both negative real numbers or 0

is this okay?

$u,v \in R^+\cup \{0\}$ and $u,v\in R^-\cup \{0\}$, not sure how to really incorporate vectors into this math expression, help is appreciated

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2 Answers 2

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How about like this: $$ u, v \in \{(x_1, x_2) \in \mathbb{R}^2: x_1 \geq 0\} \,\,\,\,\text{ or }\,\,\,\, u, v \in \{(x_1, x_2) \in \mathbb{R}^2: x_1 \leq 0\} $$ Its not too short but its clear.

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  • $\begingroup$ So is this saying that, for a component of the vectors u and v, if it's positive then the other vector component must be positive? or vice versa for negatives $\endgroup$
    – Froblinkin
    Sep 20, 2013 at 3:03
  • $\begingroup$ Not exactly. $\{(x_1, x_2) \in \mathbb{R}^2: x_1 \geq 0, x_2 \geq 0\}$ literally translates to: "all $(x_1, x_2)$ in $\mathbb{R}^2$ such that both coordinates are greater than or equal to $0$. In other words its the set of all vectors with non negative coordinates. The first part "$u, v \in$" just means they are an element of the set of vectors I just described. So the whole thing translates to $u, v$ are vectors with non negative coordinates (similar for the second line) ... which is what you wanted right? $\endgroup$
    – Pratyush Sarkar
    Sep 20, 2013 at 3:22
  • $\begingroup$ I was aiming for the x component of u and v are both >=0 or <=0 and the y component is independent of that so it can be >=0 or <=0 for both vectors when I read x1,x2 I assumed those were the same components of the vectors u,v so I could have a vector (3, -4)=u and (5,-3)=v where their components are the same sign $\endgroup$
    – Froblinkin
    Sep 20, 2013 at 3:26
  • $\begingroup$ @user2514022 I see. I'll fix my answer and see if its what you meant. $\endgroup$
    – Pratyush Sarkar
    Sep 20, 2013 at 4:22
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There is not even a standard way to denote nonnegative real numbers, see How does one denote the set of all positive real numbers?, hence I very much doubt there's a standard notation for "vectors with both components nonnegative". I would guess something like

$$ \mathbb{R}_0^+ \times \mathbb{R}_0^+ $$

but even that is not very clear. Why not just write it as

$$ u,v \in \{v | v \in \mathbb{R}^2, v_1 \cdot v_2 \ge 0\} $$

or even

$$ u,v \in \{(x,y) | x,y \in \mathbb{R}, x \cdot y \ge 0\} $$

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