Discrete Mathematics with Graph Theory (3rd Edition) 47

Discrete Mathematics with Graph Theory (3rd Edition) 47 -...

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Section 2.4 45 Transitive: If (a, b) f'V (c, d) and (c, d) f'V (e, I), then a + 2b = c + 2d and c + 2d = e + 2/, so a + 2b = e + 2/ and (a, b) f'V (e,l). The quotient set is the set of equivalence classes. We have (a, b) = {(x, y) I (x, y) f'V (a, b)} = {(x, y) I x + 2y = a + 2b} = {(x, y) I y - b = -!(x - a)} which describes the line through (a, b) with slope -!. The quotient set is the set of lines with slope -!. (b) This is an equivalence relation. Reflexive: If (a, b) E R2, then ab = ab, so (a, b) f'V (a, b). Symmetric: If (a, b) f'V (c, d), then ab = cd so cd = ab and (c, d) f'V (a, b). Transitive: If (a, b) f'V (c, d) and (c, d) f'V (e, I), then ab = cd and cd = e/, so ab = cd = e/, ab = e/ and (a, b) f'V (e, f). The quotient set is the set of equivalence classes. We have (a, b) = {(x,y) I (x,y) f'V (a,b)} = {(x,y) I xy = ab} and consider two cases. If either a = 0 or b = 0, then (a, b) = {(x, y) I xy = O}; that is, {(x, y) I x = 0 or y = O}. Hence, (a, b) is the union of the x-axis and the y-axis. On the other hand, if a # 0 and b # 0, then -- ab (a, b) = {(x,y) I xy = ab} = {(x,y) I y = -} x since x # 0 in this case. This time, (a, b) is the hyperbola whose equation is y = ab/x.
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