Answer
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Work Step by Step
⌊−x⌋ = −⌈x⌉
Let's assume x is a real number.
We know that the floor function ⌊x⌋ returns the largest integer less than or equal to x, while the ceiling function ⌈x⌉ returns the smallest integer greater than or equal to x.
Now, let's consider ⌊−x⌋ :
Since −x is the negation of x, the largest integer less than or equal to −x would be the negation of the smallest integer greater than or equal to x.
Therefore, ⌊−x⌋ = −⌈x⌉
⌈−x⌉ = −⌊x⌋
Let's assume x is a real number.
Similar to the previous case, we know that ⌊x⌋ returns the largest integer less than or equal to x, while ⌈x⌉ returns the smallest integer greater than or equal to x.
Now, let's consider ⌈−x⌉:
Since −x is the negation of x, the smallest integer greater than or equal to −x would be the negation of the largest integer less than or equal to x.
Therefore, ⌈−x⌉ = −⌊x⌋
