Surjective Ceiling and Floor Functions | Functions and Relations, Surjective Functions
Are ceiling functions and floor functions ever surjective? How would we prove it? We'll be answering those questions in today's video math lesson on surjective ceiling and floor functions!
Remember that a ceiling function takes a real number x and sends it to the smallest integer greater than or equal to x. A floor function takes a real number x and sends it to the greatest integer less than or equal to x. So the ceiling function rounds a number up, the floor function rounds a number down!
For example, ceil(2.01) = 3 and floor(2.99) = 2. The question is "are these functions ever surjective"? And indeed they can be, but whether they are or not depends entirely on the chosen domain and codomain. For some domain and codomains, a floor or ceiling function from one to the other will be surjective, and for others not. We'll see examples of both in today's lesson!
SOLUTION TO PRACTICE PROBLEM:
For our lovely function g, if it were a ceiling function from the given set to the integers, then instead of each domain element 2k - 1/2 mapping to 2k - 1, each domain element would map to 2k, because that is the smallest integer greater than or equal to 2k - 1/2. Thus, if g were a ceiling function, it would only map to even numbers and thus still not be a surjection onto the integers. But it would be surjective if the codomain was defined to be the even numbers!
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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