Why the set of integers {...-3, -2,-1,0, 1, 2, 3..) is NOT "closed" for division?

Question

Eva Abramson · Accepted Answer

When we apply division to the elements of S we get a whole slew of new numbers that are NOT in S, but rather 'outside', so S is not closed with respect to division.

To answer this question, all you need is a set of numbers (let's call it S). However, we also need an operator (divide, in this case) that can operate on any two elements in the set S. Both the numbers and the answer must be part of the set in order for it to be closed for that operation. Well, we have a problem because while #5 and 0# are both elements of S, #5/0# is undefined, and so it is not part of S. Also, #3 and 4# are both elements of S, but #3/4 and 4/3# are fractional numbers and so can't be part of S, which is a set of integers. S is not closed with respect to division because when we divide the all-integer elements of S, we obtain a large number of new numbers that are 'outside' of S rather than inside.

Genesis Allen · Answer

undefined The set of integers is not closed for division because division by zero is undefined in mathematics. If we attempt to divide any integer by zero, we encounter an undefined result, which means that the set of integers is not closed under division.