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sav08:homework10 [2008/04/30 15:57] vkuncak |
sav08:homework10 [2008/04/30 17:01] vkuncak |
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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 ===== | ||
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=== 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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=== Part c) === | === Part c) === | ||
- | 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 $\lim_{n\to\infty} f^n(x)$ when $f$ is a monotonic bounded function.) |
- | 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 $iter(0)$ (prove that the computed value is correct by definition of $iter$, that is, that the value is indeed $\sqcup$ of the set of values). 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) === | ||
Define a monotonic function $f : A \to A$ such that, for every natural number $k$, the value $iter^k(0)$ is not a fixpoint of $f$. (It may be difficult to draw $f$.) | Define a monotonic function $f : A \to A$ such that, for every natural number $k$, the value $iter^k(0)$ is not a fixpoint of $f$. (It may be difficult to draw $f$.) | ||
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