The difference of two positive numbers is not necessarily positive,
Is expressed as:
$$ (\forall x\in\Bbb Z)((x>0)\land(y>0)\rightarrow(x+y >0))$$
Is this statement correctly expressed? If so, please give a brief(or detailed) explanation.
This statement implies that if x > 0 and y > 0 then, x + y > 0, where's the necessarily part?
If the necessarily part comes from, the fact that if the R.H.S
is False then the statement is false. Then, I'm confused in the contraction with the simple analogy of another statement:
"The product of two negative numbers is always positive", expressed as:
$$ (\forall x\forall y\in\Bbb Z)((x<0) \land (y<0)\rightarrow (x*y) > 0)$$
Which is always true, so, there's no necessarily part in this at all. What I mean is, if the first statement is stated correcly, means that there's a necessarily part in this first statement, in which case, there should also be a necessarily part in the second statement(by comparison), which should not be the case, because the statement is always true.
Of course, if the first statement expressed with logic and quantifiers was wrong, everything is solved.(Note, the example the taken from Rosen's book of discrete mathematics.)