Thx Jonathan. Ok, I think I see that P and Q need to have same number of elements.

This, I think, stems from the fact that a surjection requires that every element in the set P and Q, to have its domain (or pre-image) in the A. So initially, I thought, say in your 'degenerate case', I could have {1,2,3,11} for P and {1,2,3,25,100} for Q. But then *f* and *g* would not be surjections, as there is no element in A that maps in to 3,11 or 3,25, 100.

This, I think, stems from the fact that a surjection requires that every element in the set P and Q, to have its domain (or pre-image) in the A. So initially, I thought, say in your 'degenerate case', I could have {1,2,3,11} for P and {1,2,3,25,100} for Q. But then *f* and *g* would not be surjections, as there is no element in A that maps in to 3,11 or 3,25, 100.