What does it mean if a set is "closed with respect to addition?" What about multiplication?

Question

mertsj · Answer

It means that whenever you add any two members of the set your answer is also a member os the set.

mertsj · Answer

of the set.  I meant

anonymous · Answer

Example please?

anonymous · Answer

I'll give you sets and will you tell me if they are?
{1,2}-I think no because there is no 3
{1,2,3}- think yes because 1+2=3
{1,2,3,4}-would it be because 1+2 = 3, or would it not be because 1+4 = 5

anonymous · Answer

If it's the same way for multiplication, does that mean any set with a one and something else in it is closed to multiplication?

turingtest · Answer

the set of all positive numbers is closed under addition:
whenever you add positive numbers you get another positive number as an answer
{1,2} is not closed under addition because 1+2=3 
and 3 is not a member of the set {1,2}.
it is closed under multiplication though, because 1*2=2 which is an element of the set

turingtest · Answer

(the set of positive real numbers is also closed under multiplication too)

anonymous · Answer

so it just has to have at least one number in it that is the sum or product of two other numbers in it?

turingtest · Answer

no, {1,2,3} is closed under neither addition nor multiplication, because 3+1=4 (not a member) and 2*3=6 (not a member)
so as long as there is any way to produce a member outside the set, that set is not closed under that operation.

turingtest · Answer

...no matter that 1+2=3, there exists a way to create an element outside the set, so it is not closed

anonymous · Answer

are you sure? it seems like I heard differently. But I'm unsure so I'll go with you.

turingtest · Answer

My understanding is as I said it, and quoting mertsj: "It means that whenever you add any two members of the set your answer is also a member os the set." seems like that's the same thing here's something from online: http://mathforum.org/library/drmath/view/52452.html