What does it mean for a set A to be CLOSED under addition and multiplication for a FIELD?

Answer 1

The answer was correct but here is an example

Suppose a set has (1,2,3,4,5.......)

It is closed under addition because if you add any two numbers in that set, the answer will also be in that set.

Multiplication is the same, If you multiply two numbers in that set then answer will also be in that set.


If it were (1,2,3,5,6....)
That would not work for addition because 1 plus 3 is 4 and also 2 plus 2 is 4, Therefor the set is not closed under addition

It is also not closed under multiplication because 2 times 2 is 4 and 4 is not in that set.

I hope that helps

Answer 2

There seem to be confusion in your asking this question. It seems ambiguous to me and there could be two different questions you could be asking so I am going to ask both of them because I can't tell which one do you mean.

1.What does it mean for a set to be closed under addition and multplication?
A.This is the question everyone above me has been answering. A set is said to be closed under addition if the sum of any two elements of the set also exists in the set. A set is said to be closed under multiplication if the product of any two elements of the set is also in that set.

Note:Remember that these operations (addition and multiplication) are arbitrary. I can define to them to be whatever I want. So it can be that a set is closed under one kind of addition but then not be closed in another kind of addition.

Examples:
Assuming that "natural" familiar addition and multplication
The set {1,2,3,4,...} of natural numbers is both closed under addition and multiplication.
The set [0,1] is closed under multiplication but not under addition because 0.9+0.8=1.7 is not in the set.
The set {1,2,3} is not closed under either addition nor multiplication.

2.What does it mean for a set to be closed under addition and multiplication by scalars from a field?
A.This is a different question but the only difference from the first question is that now you are not multiplying two elements of the SAME set. You are multiplying an element of the set with a scalar from a field. The addition however is the same as above.
Remember, your addition, field, and then the multiplication operation be defined. Now, your field is just as crucial. I can have a set closed using one field and then not closed using another field.

The only fields which you will probably be working with now are the real numbers and complex numbers but there are others and they could be finite or infinite.

Example:(using matrices and "familiar" operation)
The set of all 2x2 matrices with real entries is closed under addition because a real 2x2 matrix plus another real 2x2 matrix gives us a real 2x2 matrix.
The set of all real 2x2 matrices is also closed under scalar multiplication over the field of real.
If I multiply a real 2x2 matrix with a real number, I still get a real 2x2 matrix.
BUT, the set of all real 2x2 matrices is not closed under scalar multiplication if my field is complex.
A real 2x2 matrix multiplied with a complex number may or may not be real.

This sounds like you are just starting in Linear Algebra and are just being introduced to Vector Spaces. So remember, a vector space is always defined over some underlying field and the field is a lot more important than they make you think. You are probably going to work only with real vector spaces.

Answer 3

it means that the result obtained when addition and multiplication is carried out is an element of that particular field

Answer 4

It means that for a,b from the Set, the result of a+b and a*b are also elements of the Set.

Answer 5

Every element of A when added or multiplied with any other element of A, gives a result which is an element of A