A theorem of linear algebra states that if A and B are invertible matrices, then the product AB is invertible. As in Exercise 1,
(a) outline a proof of the theorem by contraposition.
(b) outline a proof of the converse of the theorem by contraposition.
(c) outline a proof of the theorem by contradiction.
(d) outline a proof of the converse of the theorem by contradiction.
(e) outline a two-part proof that A and B are invertible matrices if and only
if the product AB is invertible.
Reference:
Analyze the logical form of each of the following statements and construct just the outline of a proof by the given method. 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 proof by contraposition that if (G, *) is a cyclic group, then (G, *) is abelian.
(b) Outline a proof by contraposition that if B is a nonsingular matrix, then the determinant of B is not zero.
(c) Outline a proof by contradiction that the set of natural numbers is not finite.
(d) Outline a proof by contradiction that the multiplicative inverse of a nonzero real number x is unique.
(e) Outline a two-part proof that the inverse of the function f from A to B is a function from B to A if and only if f is one-to-one and onto B.
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