Exercise 1, § 2

Let $\tau$ be a law of composition on a set E. Denoting by F the sum set (Set Theory, II, § 4, no. 8) of E and a set $\{e\}$ consisting of a single element and identifying E and $\{e\}$ with the corresponding subsets of F, show that it is possible in one and only one way to define on F a law of composition $\overline{\tau}$ which induces on E the law $\tau$ and under which $e$ is identity element; if $\tau$ is associative, the law $\overline{\tau}$ is associative. (F is said to be derived from E "by adjoining an identity element".)