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sav07_homework_1_solution [2007/03/17 18:58] vkuncak |
sav07_homework_1_solution [2007/03/19 12:02] wikiadmin |
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| <latex> | <latex> | ||
| - | r \circ (s \cap t) \subseteq (r \circ s) \cap (r \circ t) | + | r \circ (s \cap t) \subseteq (r \circ s) \cap (r \circ t) |
| </latex> | </latex> | ||
| holds. Namely, suppose <latex>(x,z) \in r \circ (s \cap t)</latex>. | holds. Namely, suppose <latex>(x,z) \in r \circ (s \cap t)</latex>. | ||
| Then there is a y such | Then there is a y such | ||
| - | that <latex>(x,y) \in r</latex> and <latex>(y,z) \in s \cap t</latex>. Therefore, | + | that <latex>(x,y) \in r</latex> and <latex>(y,z) \in s \cap t </latex>. Therefore, |
| <latex>(y,z) \in s</latex> and <latex>(y,z) \in t</latex>. From <latex>(x,y) \in r</latex> and <latex>(y,z) \in s</latex> we | <latex>(y,z) \in s</latex> and <latex>(y,z) \in t</latex>. From <latex>(x,y) \in r</latex> and <latex>(y,z) \in s</latex> we | ||
| have <latex>(x,z) \in r \circ s</latex>. Similarly, | have <latex>(x,z) \in r \circ s</latex>. Similarly, | ||
| Line 70: | Line 70: | ||
| a precondition and 2) that if we take any other precondition, it will be a subset of | a precondition and 2) that if we take any other precondition, it will be a subset of | ||
| wp. | wp. | ||
| + | |||
| ==== Task 5 ==== | ==== Task 5 ==== | ||
| Line 79: | Line 80: | ||
| </latex> | </latex> | ||
| - | does not hold in general. | + | does not hold in general. As a counterexample, take three |
| + | distinct states s_1, s_2, and s_3, and let | ||
| + | |||
| + | <latex> | ||
| + | \begin{array}{l} | ||
| + | r = \{ (s_1,s_2), (s_1,s_3) \} \\ | ||
| + | Q_1 = \{ s_2 \} \\ | ||
| + | Q_2 = \{ s_3 \} | ||
| + | \end{array} | ||
| + | </latex> | ||
| Note: the property does hold for deterministic relations. | Note: the property does hold for deterministic relations. | ||
| Line 93: | Line 103: | ||
| is true and follows from the characterization of wp in Task 4 and the distribution of the right-hand side of implication and universal quantification | is true and follows from the characterization of wp in Task 4 and the distribution of the right-hand side of implication and universal quantification | ||
| with respect to conjunction. | with respect to conjunction. | ||
| - | |||