gmat1220 wrote:
Ian / Karishma
I may be loosing it all
But this is how I think
If xy = 0. This certainly is no brainer
x = 0 or y = 0
But the real problem is going to happen if x and y are both zero. Since x = 0 / y and since y and x are both zero we can infer that x is 0/0 or undefined. However we have assumed x=0. Hence I think we cannot assume that x and y are both zero. Can we ?
Not sure. Thoughts??
gmat1220: As per your request, let me provide the logic here.
Both, gurpreetsingh and your high school, are correct.
Given that x*y = 0.
Think of it in this way: I have two numbers. I don't know their values. But when I multiply them, I get 0. So at least one of the numbers have to be 0. Both can also be 0 since 0*0 = 0. Nothing says that they cannot be equal.
Why were you taught 'Either x or y is 0 in high school?'
Because 'Either or' implies 'At least one'. It is counter intuitive to how we think about 'Either or'. In language, we generally think 'Either or' means either A or B but not both. This is incorrect implication in logic. 'Either or' means at least one of A and B. Both are also possible. (If you do not agree, check out
https://en.wikipedia.org/wiki/Logical_disjunction)
Next, how do you explain 0 = 0/0? If you remember, in many DS questions where you have equations with denominators, you are specifically given that the denominator is not 0. e.g.
Given x = y/(x - z), x not equal to z,...
You can only divide something by x if you know that it is not zero.
xy = 0 cannot be re-written as x = 0/y because you do not know whether y is 0 or not. You do not divide an equation by a variable until and unless you know that the variable is not 0.
Hope that clears up the doubts.