EXERCISES INF3580 SPRING 2010 WEEK 12
A bird.
This document contains exercises made for INF3580. Please send any comments, errors, bug or improvement reports to this exercise set to martige@ifi.uio.no. Feedback is most welcome! Alphabetically thanks to Audun Stolpe, Espen H. Lian, Martin Giese and Rune Dahl for feedback.
The main curriculum for INF3580 spring 2010 is Semantic Web Programming by John Hebeler et al., Wiley Publishing, 2009. They have a website with additional articles and all source code used in the book at http://semwebprogramming.org/. Auxiliary curriculum is the book Foundations of Semantic Web Technologies by Hitzler, Krützsch, Rudolph, CRC Press 2009.
Keep all the work you do for these exercises in a safe
place. Setting up a version control system like cvs, svn or
git for the work you do is smart. You can create a svn repository
on IfI's svn server, see their help section for more
information. There is also a walk-through from old INF3120 on how to
set up a svn repository and connect it to Eclipse, but news is
that you'll need the plug-in subclipse to make it work. Please
contact me if you have any smart tips to share.
1 Repetition
1.1 Sets and relations
In the exercises in this section let the following be sets:
-
-
-
-
-
-
is the universal set.
1.1.1 Exercise
What is the cardinality of the sets given above?
1.1.2 Exercise
List all the elements in the following sets:
-
.
-
.
-
.
-
.
-
.
-
.
-
.
1.1.3 Exercise
Let 
and 
be two arbitrary sets and the universal
set. Draw Venn diagrams containing the sets , and
and shade the area representing the following sets:
-
.
-
.
-
.
-
.
-
.
-
.
1.1.4 Exercise
Let 
be
a relation on 
. With 
as starting point make new relations,
labelled 
to 
, by adding or removing pairs such that
they meet the following requirements:
-
is reflexive.
-
is symmetric.
-
is transitive.
-
is irreflexive.
-
, i.e., is the composition of 
and .
For each relation, list the pairs in the relation.
1.2 Semantics
Recall the notions validity, consistency from the lectures. We
say that a knowledge base 
-
is valid iff
for all interpretations/models 
of ;
-
is consistent iff for at least one
interpretation/model of .
1.2.1 Exercise
Let be a 
knowledge base 
. Answer the following
questions, and create a model to prove your answer when
appropriate.
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Is consistent?
-
Does entail
?
-
Does entail
?
-
Does there exist a model
such that 
and 
?
-
Does there exist a model
such that 
and 
?
1.2.2 Exercise
Let 
be the set of sentences
Let be an interpretation of where
-
.
-
-
-
-
This is the model is illustrated in the figure below, and is the same setup as used in the walkthrough in the lecture slides.
An interpretation of $S$.
Exercises:
-
Show that the interpretation satisfies the set of sentences
.
-
List the elements in the following sets:
-
Date: 2010-05-12 14:04:25 CEST
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