Discrete Mathematics with Graph Theory (3rd Edition) 56

Discrete Mathematics with Graph Theory (3rd Edition) 56 -...

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54 Solutions to Exercises 7. (a) Jumps in the graph o.f y = L 2x - 3 J o.ccur whenever 2x - 3 is an integer, that is, at {~ I n E Z}. (b) Jumps in the graph o.f y = rix + 7J occur whenever ix + 7 is an integer, that is, at {4n I n E Z}. (c) Jumps in the graph o.f y = L ~ J occur whenever ~(x + 3) is an integer, that is, at {x I x = 5n-3,n E Z}. 8. (a) [BB] The answer is ''yes.''· By definition o.f "flo.o.r", we kno.w that kx is the unique integer satisfy- ing n - 1 < kx < n. Thus !! - 1 .( x < !!. Since !! - 1 < !! - 1 we have !! - 1 < x < !!. - k k - k· k - k k' k - k Thus x = LIJ. . (b) The multiples o.f 3 in the indicated interval are 3,6,9, ... ,3k, where 3k = L 50000 J. Thus k = L 50~OO J = 16666. 9. [BB] We kno.w that Lx J is an integer less than o.r equal to. x and that Ly J is an integer less than o.r equal to. y. Thus LxJ + LyJ is an integer less than o.r equal to. x + y. Since Lx + yJ is the greatest such integer, we get the desired result. 10. We kno.w that
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