Theorem 4.10: a) If n is any natural number,

(n + 1) - 1 = n

b) If n is any nonzero natural number,

(n - 1) + 1= n

Proof: a) By Definition 4.3,

So the largest element of n + 1 is n. By Definition 4.7,

(n - 1) + 1= n.

b) The predecessor of n is

n - 1 = { 0, 1, . . . , ((n - 1) - 1}

The successor of n - 1 is

= {0, 1, . . . , n - 1} = n.