Prove that the floor(2x)=floor(x)+floor(x+1/2)i need the proof with cases.
I merely desire to understand if the atheist can certainly supply the the wonderful option answer as they're so beneficial that God did no longer create life? - i'm able to instruct that the biblical version is a myth. All of genesis became plagiarized from the Sumerian pills having no longer something to do with deities so god starting to be life is a myth. - As to proving the place life got here from, we settle for a organic technique when you consider that that has info. A deity magically forming life is a myth when you consider which you have yet to instruct a deity exists and that what youi use as a info is a properly shown myth. - There you have surely info and info and no hypothesis.
Suppose that we split up x into its integer part y and its decimal part z (for example, we can split 3.6 up into 3 and 0.6). Then, we have: x = y + z. Then: floor(2x) = floor(2y + 2z), floor(x) = floor(y + z), and floor(x + 1/2) = floor(y + z + 1/2). For 0 < z < 1, floor(y + z) = y for all integers y (y is defined to be an integer, so this is fine). As for floor(2y + 2z) and floor(y + z + 1/2), we need to take cases. Now, if 0 < z < 0.5, then 0 < 2z < 1 and 0 < z + 1/2 < 1 and so: 2y < 2y + 2z < 2y + 1 and y < y + z + 1/2 < y + 1, which implies that 2x is some number between 2y and 2y + 1 and that x + 1/2 is some number between y and y + 1, which implies: floor(2x) = x and floor(x + 1/2) = x, so floor(2x) = floor(x) + floor(x + 1/2) when x is some integer plus a decimal between 0 and 0.5. As for the case 0.5 < z < 1, 1 < 2z < 2 and 1 < z + 1/2 < 1.5, so: 2y + 1 < 2y + 2z < 2y + 2 and y + 1 < y + z + 1/2 < y + 1.5, yielding floor(2y + 2z) = 2y + 1, floor(y + z + 1/2) = y + 1, and floor(y + z) = y, and so: floor(2y + 2z) = floor(y + z + 1/2) + floor(y + z) == floor(2x) = floor(x) + floor(x + 1/2). Combining these two cases proves floor(2x) = floor(x) + floor(x + 1/2) when x is not an integer. To prove the case when x is an integer, notice that: floor(2x) = 2x, floor(x) = x, and floor(x + 1/2) = x, when x is an integer, which shows that floor(2x) = floor(x) + floor(x + 1/2) when x is an integer. Combining these three cases yields floor(2x) = floor(x) + floor(x + 1/2) for all x (as required). I hope this helps!
