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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 ===== | ||
| 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 ===== | ||
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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 $iter(0)$ (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? | ||