If `[.]` denotes the greatest integer function and `x,yepsilonR,"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]`

Options

A. `[x+y]ge[x]+[y]`
B. `[x+y]le[x]+[y]`
C. `[([x])/n]=[x/n]`
D. `[x+1/2]=[2x]-[x]`

Answer

A, C, D

Explanation

(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`