 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2007/b/5/ii/solution

Induct on

n

. The initial equation is

0 + 0 = 0

by the zero clause. If

0 + n = n

, the successor clause gives

0 + n^{+} = (0 + n)^{+} = n^{+} .

Consequently

0 + n = n (n \in ω) .

This proves the left-identity property independently of the right-identity equation built into the recursive definition.

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