sukhman wrote:If there are more than two numbers in certain list, is each of the numbers in the list equal to 0?
1). The product of any two numbers in the list equal to 0.
2). The sum of any two numbers in the list equal to 0.
I like this: it's a great question. I'm happy to help.
Statement #1:
The product of any two numbers in the list equal to 0
Here, every element could be zero, but we also could have a list {0, 0, 0, 1, 0, 0}. If just one of the numbers is non-zero, then it's product with anything else would still be zero. Beginning with this statement, we could construct either a "yes" or "no" answer to the prompt question. This statement, alone and by itself, is
not sufficient.
Statement #2: The sum of any two numbers in the list equal to 0.
Well, one possibility is a list in which each element is zero. Is it possible to have a list with nonzero elements that satisfies this condition?
Suppose we have {-2, 2, 0, 0} --- well, (-2) + 2 = 0, but 2 + 0 = 2, so this doesn't meet the requirement.
What about {-2, 2, -2, 2) --- again, (-2) + 2 = 0, but 2 + 2 = 4. Again, this doesn't meet the requirement.
If the set could have just two elements, then we could have {-2, 2}, but as soon as we have another element, either it's zero (doesn't have a zero sum with either), or positive (doesn't have a zero sum with +2) or negative (doesn't have a zero sum with -2). Thus, it's absolutely impossible to have a set with more than two elements that meets this condition unless all the elements individually equal 0. This statement gives us a definitive "yes" answer to the prompt question, so this statement, alone and by itself, is
sufficient.
Answer = [spoiler]
(B)[/spoiler]
Does all this make sense?
Mike
