When discussing set operations, what does it mean if a set is closed under an operation?

A set being closed under an operation means that when you apply that operation to any two elements within the set, the result is also an element of that same set. This is a fundamental concept in set theory and is often illustrated in various mathematical contexts such as addition, multiplication, and other operations.

For example, consider the set of even numbers. If you take any two even numbers and add them together, the result is always an even number, which means the set of even numbers is closed under addition. This characteristic is crucial in defining algebraic structures like groups, rings, and fields.

The other options do not accurately capture the meaning of "closed under an operation." The idea of always yielding zero, never producing a number greater than the set's elements, or being restricted to prime numbers does not reflect the foundational principle of closure in set operations. Instead, closure specifically refers to whether the result of the operation remains within the original set, which is why the correct answer is the first option.