Mathematical Notation for Conditional Statements

Question

In computer science, we would write a condtional statement like so:

if (cross_product(vec_a, vec_b) = 0) 
   parallel = true;

But if we wanted to represent something like this mathematically, then what would be the correct notation?

Answer 1

You would usually use implications and arrows for it, e.g.

cross_product(..) = 0 $\Rightarrow$ parallel = true

to say that one follows from the other.

Answer 2

You could use an "if-then" statement. So, for example,

If $a \times b = 0$, then $a$ and $b$ are parallel.

But so far, this is still an incomplete mathematical statement, because we haven't explained what $a$ and $b$ are. So, a more complete statement would be

For every pair of vectors $a,b \in \Bbb{R}^3$, if $a \times b = 0$ then $a$ and $b$ are parallel.

There might be some ambiguity about what "parallel" means based on who you ask and how you define it (eg. is $(1,0,0)$ parallel to $(0,0,0)$?). So, to remove this potential ambiguity, we might prefer to use a more agreed upon terminology, which is that of linear dependence (two vectors are linearly dependent if and only if one of them is a scalar multiple of the other):

For every pair of vectors $a,b \in \Bbb{R}^3$, if $a \times b = 0$ then $a$ and $b$ are linearly dependent.

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I prefer to write full English sentences as opposed to using logical symbols only. However, if you want to, you could say the following:

$\forall a,b \in \Bbb{R}^3,$ $a \times b = 0 \implies$ $a$ and $b$ are linearly dependent.