What is the smallest closed subset of Z containing 2 and 0?

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In summary, the conversation discusses finding the smallest subset A of Z that is closed with respect to addition and contains the element 2. The answer given was the set of all positive even integers, but it is not correct according to the definition of closure. The correct set would be A = {0, 2} as it satisfies the definition, unlike the set of all positive even integers.
  • #1
Gear300
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The question given is: Determine the smallest subset A of Z such that 2 ε A and A is closed with respect to addition.

The answer given was the set of all positive even integers, but I was thinking that the smallest subset would be the given element and the identity element (0 in this case) so that A = {0,2}...wouldn't this be more accurate?
 
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To understand why that is not correct, you have to know what closure of an operator over a set means.

A set S is closed under a binary operator + iff for all x, y in S, x + y is in S.

A = {0, 2} isn't closed under addition because the definition is not satisfied. We can find a counter example where x and y are both in S, but x + y is not. The counter example is with x = 2 and y = 2.
 
  • #3
Closed means you can do it to any two (possibly non-unique) elements and get an answer in your set.

S = {0, 2} doesn't work since 2+2=4 is not in S.
 
  • #4
I see. Thanks...funny I didn't notice that.
 

1. What is a "closed operation"?

A closed operation is a mathematical term that refers to an operation (such as addition, subtraction, multiplication, or division) that, when applied to any two elements from a given set, always produces a result that is also a member of that same set. In other words, the result of the operation is "closed" within the original set.

2. What does the notation "A = {0,2}" mean?

The notation "A = {0,2}" means that the set A contains the elements 0 and 2. In other words, A is a set that contains only the numbers 0 and 2.

3. How do you know if an operation is closed on a particular set?

To determine if an operation is closed on a set, you can apply the operation to any two elements from the set and check if the resulting value is also a member of the set. If it is, then the operation is closed on that set. In the case of "A = {0,2}", we can see that adding 0 and 2 together gives us 2, which is also a member of the set A. Therefore, addition is closed on A.

4. What are the benefits of a closed operation?

Closed operations are useful in mathematics because they allow for consistent results and make it easier to perform calculations. They also help to ensure that the resulting values are within the original set, which can be important in certain applications.

5. Can an operation be closed on one set but not on another?

Yes, an operation can be closed on one set but not on another. For example, addition may be closed on "A = {0,2}" but not on "B = {1,3}". This is because when we add 1 and 3, we get 4, which is not a member of the set B. Therefore, it is important to specify which set an operation is being performed on when discussing closed operations.

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