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Let S be equivalence relation defined on $\{x : x ∈ \mathbb{R},\ 0 ≤ x ≤ 5\} $defined by $xSy$ if and only if $[x] = [y]$. What are the equivalence classes of $S$?

Note: $[q]$ is defined to be the smallest integer greater than or equal to $q$. You can think of it as “$q$ rounded up”. You don’t need to prove that $S$ is an equivalence relation.

My answer is as follows but i am not sure if this is what they are looking for:

Equivalence classes of $S$ = $\{[0, 1), [1,2), [2,3), [3,4), [4,5)\}$

mlc
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Dave Kumar
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    the smallest integer greater than or equal to q Then $\lceil 0 \rceil = 0$ while $\lceil 0.5 \rceil = 1,$, so $0$ and $0.5$ do not belong into the same equivalence class $[0,1),$. – dxiv Apr 24 '17 at 05:48
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    Your equivalence classes don't cover all elements of $S$, – celtschk Apr 24 '17 at 05:48

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You are almost there. The interval should be closed on the right, and you left out $5$. The equivalence classes are: $$S= \left\{ {0}, (0,1], (1,2], (2,3], (3,4], (4,5] \right\}$$

mlc
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