A binary operation on set has to fulfill two conditions given below:
1) exactly one element is assigned to each possible ordered pair of element of ,
2) for each ordered pair of elements of , the element assigned to it is again in .
Now there is an example in Fraleigh's abstract algebra book(Example 2.25):
Let be a set consisting of 20 people, no two of whom are of the same height and let , where is the shortest person in who
is taller than both and . This is not everywhere defined, since if either or is the tallest person in the set, is not
Now in the exercise there is a problem(exercise 20):
On , define by letting , where is the smallest integer greater than both and .
The solution manual says this is a valid binary operation. Why these two same problems with different wording have different outcomes?
For exercise 20 my problem is:
There can be a positive integer or that is greater than . In that case shouldn't this binary operator be undefined according to example
Then solution manual says that exercise 20 is a valid binary operation.
Is it possible to tell me why in exercise 20 the binary operation is valid when example 2.25 says that the binary operation is invalid?