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Closure of a set of even numbers | TOMATO objective 27

Try this beautiful problem from TOMATO Objective no. 27 based on Closure of a set of even numbers.

Problem: Closure of a set of even numbers

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.

Answer: (D) [1], [2], [3]

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