Floors and ceilings. For all proofs in this question, you may use the following fact Vx E R,0

Floors and ceilings. For all proofs in this question, you may use the following fact<br>Vx E R,0 < x - [x] < 1,<br>(1)<br>and its generalization to the ceiling function:<br>Vx E R,0 < [x] – x < 1.<br>(2)<br>x +1<br>(a)<br>Use proof by cases to prove that for all x E Z,<br>2<br>(b)<br>the exact statement you are attempting to prove (either the original statement or its negation).<br>Prove or disprove each of the following. In each case, first write down in symbolic notation<br>i. Vx, y E R, x <y= [x] < [y]<br>ii. Vx, y E R, [x] < [y] = x <y<br>

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Jun 04, 2022

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Floors and ceilings. For all proofs in this question, you may use the following fact Vx E R,0

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