definition of floor map
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of map.
Target Context
- The reader will have a definition of floor map.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*fl\): \(: [0, \infty) \to \mathbb{N}, r \mapsto Max (\{n \in \mathbb{N} \vert n \le r\})\)
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Conditions:
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2: Note
Let us see that \(fl\) satisfies some properties.
1): For each \(r \in [0, \infty)\), \(fl (r) \le r\), and if and only if \(r \in \mathbb{N}\), \(fl (r) = r\), because \(r = n + r'\) where \(0 \le r' \lt 1\), and \(fl (r) = fl (n + r') = n\), and \(fl (r) = n \le n + r' = r\); if \(r \in \mathbb{N}\), \(r' = 0\), and \(fl (r) = n = r\), and if \(fl (r) = r\), \(n = r\), so, \(r \in \mathbb{N}\).
2): For each \(r \in [0, \infty)\), \(fl \circ fl (r) = fl (r)\), because \(fl (r) \in \mathbb{N}\) and 1) applies.
3): For each \(r, r' \in [0, \infty)\) such that \(r \le r'\), \(fl (r) \le fl (r')\) but not necessarily \(r \lt r'\) implies \(fl (r) \lt fl (r')\), because if \(r \le r'\), \(\{n \in \mathbb{N} \vert n \le r\} \subseteq \{n \in \mathbb{N} \vert n \le r'\}\) and the proposition that for any partially-ordered set, any subset, and any subset of the subset, if the infimum of the subset and the infimum of the subset of the subset exist, the infimum of the subset is equal to or smaller than the infimum of the subset of the subset, and if the supremum of the subset and the supremum of the subset of the subset exist, the supremum of the subset is equal to or larger than the supremum of the subset of the subset applies: the maximums are the supremums; if \(r \lt r'\), if \(r = 1.1\) and \(r' = 1.2\), \(fl (r) = 1 = fl (r')\), so, "\(fl (r) \lt fl (r')\)" does not necessarily hold.
4): For each \(r \in [0, \infty)\), \(r \lt fl (r + 1)\), because \(r = n + r'\) where \(0 \le r' \lt 1\), and \(fl (r + 1) = fl (n + 1 + r') = n + 1\) and \(n + r' \lt n + 1\).
5): For each \(r, r' \in [0, \infty)\), if and only if \(fl (r) \le r'\), \(r \lt fl (r' + 1)\), because if \(fl (r) \le r'\), if \(fl (r' + 1) \le r\), \(fl (r' + 1) = fl \circ fl (r' + 1) \le fl (r)\), by 2) and 3), \(\le r'\), a contradiction against 4); if \(r \lt fl (r' + 1)\), \(fl (r) \le r \lt fl (r' + 1)\), by 1), but \(r' = n + r''\) where \(0 \le r'' \lt 1\), and \(fl (r' + 1) = fl (n + 1 + r'') = n + 1\), so, \(fl (r) \lt n + 1\), but as \(fl (r) \in \mathbb{N}\), \(fl (r) \le n \lt n + 1\), and \(fl (r) \le n \le n + r'' = r'\).
