Chapter 1.4, Problem 1E

Analyze the logical form of each of the following statements and construct just the outline of a proof. Since the statements may contain terms with which you are not familiar, you should not (and perhaps could not) provide any details of the proof.

1. Outline a direct proof that if $\left(G,*\right)$ is a cyclic group, then $\left(G,*\right)$ is abelian.
2. Outline a direct proof that if B is a nonsingular matrix, then the determinant of B is not zero.
3. Suppose A, B, and C are sets. Outline a direct proof that if A is a subset of B and B is a subset of C, then A is a subset of C.
4. Outline a direct proof that if the maximum value of the differentiable function f on the closed interval $\left[a,b\right]$ occurs at $x_0,$ then either $x_0=a$ or $x_0=b$ or $f\text{'}\left(x_0\right)=0.$
5. Outline a direct proof that if A is a diagonal matrix, then A is invertible whenever all its diagonal entries are nonzero.