C.2 Example Self-Explanations

Theorem.No odd integer can be expressed as the sum of three even integers.

Proof.

(L1)

Assume, to the contrary, that there is an odd integer $x$, such that $x=a+b+c$, where $a,b,$ and $c$ are even integers.

(L2)

Then $a=2k,b=2l,$ and $c=2p$, for some integers $k,l$, and $p$.

(L3)

Thus $x=a+b+c=2k+2l+2p=2(k+l+p)$.

(L4)

It follows that $x$ is even; a contradiction.

(L5)

Thus no odd integer can be expressed as the sum of three integers. $\square$

After reading this proof, one reader made the following self-explanations:

• •

  ‘‘This proof uses the technique of proof by contradiction.’’

• •

  ‘‘Since $a,b$ and $c$ are even integers, we have to use the definition of an even integer, which is used in L2.’’

• •

  ‘‘The proof then replaces $a,b$ and $c$ with their respective definitions in the formula for $x$.’’

• •

  ‘‘The formula for $x$ is then simplified and is shown to satisfy the definition of an even integer also; a contradiction.’’

• •

  ‘‘Therefore, no odd integer can be expressed as the sum of three even integers.’’