Question 5.6: Show that ≤m is a transitive relation.

Suppose A \leq _{m} B and B \leq _{m} C. Then there are computable functions f and g such that x\in A\Longleftrightarrow f(x) \in B and y\in B\Longleftrightarrow g(y) \in C. Consider the composition function h (x)= g(f(x)). We can build a TM that computes h as follows: First, simulate a TM for f (such a TM exists because we assumed that f is computable) on input x and call the output y. Then simulate a TM for g on y.
The output is h (x)= g(f(x)). Therefore, h is a computable function. Moreover, x\in A \Longleftrightarrow h(x)\in C. Hence A \leq _{m} C via the reduction function h.