> Hi, > > I'm having trouble understanding "the least integer > greater than or equal to z" in the following > question: > > 96. For all Z, [z] denotes the least integer greater > than or equal to z. Is [x] = 0? > > s1) -1 < x < -0.1 > s2) [x + 0.5] = 1 > > What does "the least integer greater than or equal to > z" mean? Can someone explain it to me in English? > >
Consider two cases:
1. z is an integer. Let z=n Then n, n+1, n+2, .... are all integers >= n And the least of these is n So "the least integer >= z" is n. That is [z]=n
2. z is not an integer. Let z=n+d where 0<d<1 Then n+1, n+2, n+2, .... are all integers >= n And the least of these is n+1 So "the least integer >= z" is n+1. That is [z]=n+1
Q 96. asks if [x]=0 in two cases:
s1) -1<x<-0.1 In this case x=-1+d where 0<d<0.9 So [x] = -1+1 =0 and the answer is Yes.
s2) [x+0.5] = 1 So 0<(x+0.5)<=1 So -0.5<x<=0.5 So the answer is not Yes for the whole range, so it must be No. More specifically: For -0.5<x<=0 the answer is Yes, but for 0<x<=0.5 the answer is No.