Solve the following equation for \(x\text{,}\) first assuming that the coefficients belong to \(\Z\text{,}\) and then assuming the coefficients belong to \(\Z_5\) (see the tables on the Operation: Properties in-class activity).
For each of the combinations of properties given below, create a new operation on the set \(\{1, 2, 3\}\) by giving its operation table. Briefly explain why your example has (or doesn’t have) the properties requested.
Is there an operation on a set of three elements (such as the set \(\{1,2,3\}\)) that is associative, has an identity, and for which every element has an inverse, but which is not commutative? Find one or explain why not.
Bonus: Generalize the previous question as best you can. Are there operations on larger sets of elements that are not commutative even though they are associative, have an identity, and for which every element has an inverse? What sizes could this work for?
