Analyze the logical form of each of the following statements and construct just the outl...

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.

(a) Outline a direct proof that if (G, *) is a cyclic group, then (G, *) is abelian.

(b) Outline a direct proof that if B is a nonsingular matrix, then the determinant of B is not zero.

(c) 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.

(d) Outline a direct proof that if the maximum value of the differentiable function on the closed interval [a, b] occurs at then either

(e) Outline a direct proof that if A is a diagonal matrix, then A is invertible whenever all its diagonal entries are nonzero.