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Problem:

S is the set whose elements are zero and all even integers, positive and negative. Consider the 5 operations- [1] addition;  [2] subtraction;   [3] multiplication; [4] division; and [5] finding the Arithmetic Mean. Which of these operations applied to any pair of elements of S, yield only elements of S.

Solution:

Adding, subtracting and multiplying even numbers  yield even numbers.

Division may yield non-integer ($\frac {2}{4} = \frac{1}{2}$ ) or odd number ($\frac {12}{4} = 3$) both of which are not members of S. So division is ruled out.

Finally arithmetic of two even numbers may be a odd number. For example, $\frac { 4 + 6}{2} = 5$.

Thus operations [1], [2], [3] work.