Show that the following statements are false (that is they are not equal):
(a) $⌊2 × x⌋ = 2 × ⌊x⌋$
(b) $⌈2 × x⌉ = 2 × ⌈x⌉$
(c) $⌈x⌉ = ⌊x⌋ + 1$
We will show the statements are false by providing a counter-example.
(a) Choosing $x=0.5$
$\lfloor 2 \times 0.5 \rfloor = \lfloor 1 \rfloor = 1$
$2 \times \lfloor 0.5 \rfloor = 2 \times 0 = 0$
(b) Choosing $x=0.5$
$\lceil 2 \times 0.5 \rceil = \lceil 1 \rceil = 1$
$2 \times \lceil 0.5 \rceil = 2 \times 1 = 2$
(c) Choosing $x=1$
$\lceil 1 \rceil = 1$
$\lfloor 1 \rfloor + 1 = 1 + 1 = 2$