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sav08:homework10 [2008/04/30 15:57] vkuncak |
sav08:homework10 [2008/04/30 16:10] piskac |
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| r = \{ (F_1,F_2) \mid \models F_1 \rightarrow F_2 \} | r = \{ (F_1,F_2) \mid \models F_1 \rightarrow F_2 \} | ||
| \] | \] | ||
| - | Check whether $r$ is reflexive, symmetric, and transitive relation. | + | Check whether $r$ is reflexive, antisymmetric, and transitive relation. |
| ===== Problem 2 ===== | ===== Problem 2 ===== | ||
| Line 29: | Line 29: | ||
| === Part a) === | === Part a) === | ||
| - | Prove that $f$ is monotonic. | + | Prove that $f$ is monotonic and injective (so it is strictly monotonic). |
| === Part b) === | === Part b) === | ||
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| Define $iter(x) = \sqcup \{ f^n(x) \mid n \in \{0,1,2,\ldots \}\}$. (This is in fact equal to $\lim_{n\to\infty} f^n(x)$.) | Define $iter(x) = \sqcup \{ f^n(x) \mid n \in \{0,1,2,\ldots \}\}$. (This is in fact equal to $\lim_{n\to\infty} f^n(x)$.) | ||
| - | Compute $iter(0)$. Is $iter(0)$ a fixpoint of $f$? Is $iter(iter(0))$ a fixpoint of $f$? Is $f$ an $\omega$-continuous function? | + | Compute $iter(0)$ (if you can, prove that the computed value is correct by definition of $iter$). Is $iter(0)$ a fixpoint of $f$? Is $iter(iter(0))$ a fixpoint of $f$? Is $f$ an $\omega$-continuous function? |
| === Optional part d) === | === Optional part d) === | ||