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If `[.]` denotes the greatest integer function and `x,yepsilonRr,"n" epsilonN` then which of the following is true?A. `[x+y]ge[x]+[y]`B. `[x+y]le[x]+[y]`C. `[([x])/n]=[x/n]`D. `[x+1/2]=[2x]-[x]` |
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Answer» Correct Answer - A::C::D (A) & (B) `x+y=[x]+[y]+alpha+beta` where `alpha={x},beta={y}` `:.0ltalpha+betalt2implies[alpha+beta]=0` or `1` `:.[x+y]ge[x]+[y]` (C) By division a lgorithm, `[x]=qn+r(0lerlen-1)` Let `x=[x]+alpha,0ltalphalt1` Then `x=1n+ralpha` `:.[x/n]=q ( :.0ltr+alphaltn)` (D) let `x=n+alpha "n" epsilonN` Consider Case-1 `0lealphale1/2` Case II `1/2lealphale1` |
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