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modular:ordered_sets [2014/01/30 00:22] marje created |
modular:ordered_sets [2014/01/31 01:07] marje |
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| - $\mathbb Z$, $\mathbb N$, or any other subset of the set of all reals $\mathbb R$ with the usual "less than or equal" relation $\le$ (or "greater than or equal" $\ge$), | - $\mathbb Z$, $\mathbb N$, or any other subset of the set of all reals $\mathbb R$ with the usual "less than or equal" relation $\le$ (or "greater than or equal" $\ge$), | ||
| - any collection of sets with the inclusion relation $\subset$, e.g., $\{\{1\}, \{1,2\}, \{3\}, \emptyset\}$. | - any collection of sets with the inclusion relation $\subset$, e.g., $\{\{1\}, \{1,2\}, \{3\}, \emptyset\}$. | ||
| - | - $\mathbb N$ with the division relation $\mid$ (or with the division relation $\divby$). | + | - $\mathbb N$ with the division relation $\mid$. |
| Note that $\mathbb Z$ with the division relation is not a poset as this relation is not antisymmetric on $\mathbb Z$ (because, e.g., $1 \mid -1$ and $-1 \mid 1$ but $1 \neq -1$). ((But this relation is a [[http://en.wikipedia.org/wiki/Preorder|preorder]] on $\mathbb Z$.)) | Note that $\mathbb Z$ with the division relation is not a poset as this relation is not antisymmetric on $\mathbb Z$ (because, e.g., $1 \mid -1$ and $-1 \mid 1$ but $1 \neq -1$). ((But this relation is a [[http://en.wikipedia.org/wiki/Preorder|preorder]] on $\mathbb Z$.)) | ||
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| Of the examples above, sets in 1. are totally ordered, while the set in 3. is not, because, e.g., $2$ and $3$ are incomparable (neither $2 \mid 3$ nor $3 \mid 2$). | Of the examples above, sets in 1. are totally ordered, while the set in 3. is not, because, e.g., $2$ and $3$ are incomparable (neither $2 \mid 3$ nor $3 \mid 2$). | ||
| - | <WRAP indent> | + | <WRAP task> |
| - | :?: Which of the following sets are totally ordered? | + | Which of the following sets are totally ordered? |
| - The set on the Hasse diagram above. ++Answer| No, because, e.g., $\{1\}$ and $\{3\}$ are incomparable.++ | - The set on the Hasse diagram above. ++Answer| No, because, e.g., $\{1\}$ and $\{3\}$ are incomparable.++ | ||
| - The set given by the following ++diagram. | | - The set given by the following ++diagram. | | ||
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| The symmetrical notion, the **greatest lower bound** (aka **infimum**) of $A$ is denoted $\inf A$. | The symmetrical notion, the **greatest lower bound** (aka **infimum**) of $A$ is denoted $\inf A$. | ||
| </box> | </box> | ||
| - | |||
| - | <WRAP indent> | ||
| - | :?: | ||
| - | Exercises on suprema. TODO | ||
| - | </WRAP> | ||
| <box 100% round blue> | <box 100% round blue> | ||
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| </box> | </box> | ||
| - | <WRAP indent> | + | |
| - | :?: | + | ----------------------------------------------------------------------------------------------------------------------------------------- |
| - | Which of the following sets are lattices? | + | |
| - | </WRAP> | + | |