Discrete Mathematics with Graph Theory (3rd Edition) 46

Discrete Mathematics with Graph Theory (3rd Edition) 46 -...

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44 Solutions to Exercises 16. (a) [BB] The given statement is an implication which concludes "x - y = x - y," whereas what is required is a logical argument which concludes "so rv is reflexive." A correct argument is this: For any (x,y) E R2, x - Y = x - y; thus, (x, y) rv (x, y). Therefore, rv is reflexive. (b) There is confusion between the elements of a binary relation on a set A (which are ordered pairs) and the elements of A which are themselves ordered pairs in this situation. The given statement is correct provided each of x and y is understood to be an ordered pair of real numbers, and we understand n = {(x, y) I x rv y} but this is very misleading. Much better is to state symmetry like this: if (x, y) rv (u, v), then (u, v) rv (x, y). (c) The first statement asserts the implication "x - y = u - v ~ (x,y) rv (u,v)" which is the converse of what should have been said. Here is the correct argument: If (x, y) rv (u, v), then x - y = u - v, so-u - v = x - y and, hence, (u, v) rv (x, y). (d) This suggested answer is utterly confusing. Logical arguments consist of a sequence of implica-
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