Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
sav08:homework10 [2008/04/30 16:56] vkuncak old revision restored |
sav08:homework10 [2008/04/30 17:01] vkuncak |
||
---|---|---|---|
Line 7: | Line 7: | ||
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 ===== | ||
Let $(A,\sqsubseteq)$ be a [[:partial order]] such that every set $S \subseteq A$ has the greatest lower bound. Prove that then every set $S \subseteq A$ has the least upper bound. | Let $(A,\sqsubseteq)$ be a [[:partial order]] such that every set $S \subseteq A$ has the greatest lower bound. Prove that then every set $S \subseteq A$ has the least upper bound. | ||
+ | |||
===== Problem 3 ===== | ===== Problem 3 ===== | ||
Line 36: | Line 37: | ||
=== 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 $iter(0)$ (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? | 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? |