What does it mean for a set to be "closed"?

Question

1 An expression belonging to a closed set cannot be used to create an expression that is not of that set.

2 Members of a closed set cannot be added, subtracted, multiplied, or divided.

3 An operation performed on members of that set will result in a member of the same set.

4 The set has a finite number of members, as opposed to being infinitely large.

Answer 1

The correct interpretation of what it means for a set to be "closed" is:

3. An operation performed on members of that set will result in a member of the same set.

In mathematical terms, a set is said to be closed under a particular operation (such as addition, subtraction, multiplication, or division) if performing that operation on members of the set always yields a result that is also a member of the set.

For example:

• The set of even integers is closed under addition, because the sum of any two even integers is also an even integer.
• The set of natural numbers is not closed under subtraction, because subtracting two natural numbers can yield a negative number, which is not a natural number.

The other statements do not accurately describe the concept of a closed set in this context.