How to Think Like a Mathematician : A Companion to Undergraduate Mathematics (9781139738071) by Houston Kevin

Author:Houston, Kevin
Language: eng
Format: epub
Publisher: Cambridge Univ Pr

For the converse, as a, b, c and d are not all zero, we can assume without loss of generality that a is non-zero. Hence, x = (p − by)/a is a solution to (1) for every y . Now, assuming (1) has this solution, and because ad − bc = 0 we have

Hence, if cp = aq, then a solution to (1) gives a solution to (2), and therefore the system has an infinite number of solutions.

On the other hand, if cp = aq, then a solution to (1) is not a solution to (2), and therefore no solutions exist.

(iii)

Consider the statement, also from Chapter 16 exercises: For sets A and B we have A B if and only if (A\B) (B\A) .

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