7. Explain why the following data are not sufficient for obtaining a group and explain in which cases you may complete the given data to obtain a group; otherwise, explain why no group could result from the given data.

(a) Consider the set of finite binary strings with the operation $x \circ y = x$.

(b) Consider the set of injective functions $f: \mathbb{N} \rightarrow \mathbb{N}$, that is, functions satisfying

$f(x) = f(y)$ implies $x = y$ for all $x, y \in \mathbb{N}$.

Let e be the function $e(n) \stackrel{def}{=} n$ for all $n \in \mathbb{N}$. Define $f \circ g$ to be the function that maps $n$ to $f(g(n))$ for all $n \in \mathbb{N}$.

(c) Consider the set of finite binary strings, let e be the empty string of length 0, and define $x \circ y$ to be the bitwise merge of $x$ and $y$. For example, $a_1a_2a_3a_4a_5 \circ b_1b_2b_3$

equals $a_1b_1a_2b_2a_3b_3a_4a_5$.