Let the sets S = {(n, m) ∈ N × N: n + m = 10} and π1 (S) = {n:
there exists m ∈ N such that (n, m) ∈ S}. (Note that π1 (S) ⊆ N.)
(a) Show that 8 ∈ π1 (S) and give an example of an element x
such that x ∈ N ∖ π1 (S).
(b) Show that π1 (S) = N≤10. (Remember that N≤10 is defined
as {n ∈ N: n ≤ 10}.)
(c) Show that, given the sets A, B ⊆ N, S ≠ A × B. (Help:
can reason by contradiction.)