Which of these statements about the floor and ceiling functions are correct?
Statement I : \(\left\lfloor {2x} \right\rfloor = \left\lfloor x \right\rfloor + \left\lfloor {x + (1/2)} \right\rfloor \) for all real number x
Statement II : \(\left\lceil {x + y} \right\rceil = \left\lceil x \right\rceil + \left\lceil y \right\rceil \) for all real numbers x and y
1.Both Statement I and Statement II are true
2.Both Statement I and Statement II are false
3.Statement I is true but Statement II is false ✓ Correct
4.Statement I is false but Statement II is true
Solution
Given:
Statement 1:
\(\left\lfloor {2x} \right\rfloor = \left\lfloor x \right\rfloor + \left\lfloor {x + (1/2)} \right\rfloor \)
Real number: All Positive numbers except zero.
suppose x = 1,2,3----
Take x = 1
\(\left\lfloor {2\times 1} \right\rfloor = \left\lfloor 1 \right\rfloor + \left\lfloor {1 + (1/2)} \right\rfloor \)
\(\left\lfloor {2} \right\rfloor = \left\lfloor 1 \right\rfloor + \left\lfloor { (3/2)} \right\rfloor \)
\(\left\lfloor {2} \right\rfloor = \left\lfloor 1 \right\rfloor + \left\lfloor { 1.5} \right\rfloor \)
2 = 1 + 1
2 = 2
Statement 2:
\(\left\lceil {x + y} \right\rceil = \left\lceil x \right\rceil + \left\lceil y \right\rceil \)
suppose x = 1.2 and y = 8.6
\(\left\lceil {1.2 + 8.6} \right\rceil = \left\lceil 1.2 \right\rceil + \left\lceil 8.6 \right\rceil \)
\(\left\lceil {9.8} \right\rceil = \left\lceil 1.2 \right\rceil + \left\lceil 8.6 \right\rceil \)
10 = 2 + 9
10 =! 11
It shows the Above Statements is false for x,y belongs to Real numbers.
The Correct Option is 1