INF3580 - V?r 2010

EXERCISES INF3580 SPRING 2010 WEEK 12

A bird.

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. .
  2. .
  3. .
  4. .
  5. .
  6. .
  7. .

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. .
  2. .
  3. .
  4. .
  5. .
  6. .

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:

  1. is reflexive.
  2. is symmetric.
  3. is transitive.
  4. is irreflexive.
  5. , 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.

  1. Is consistent?
  2. Does entail ?
  3. Does entail ?
  4. Does there exist a model such that and ?
  5. 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.

An interpretation of $S$.

Exercises:

  1. Show that the interpretation satisfies the set of sentences .
  2. List the elements in the following sets:

Author: Martin G. Skj?veland <martige@ifi.uio.no>

Date: 2010-05-12 14:04:25 CEST

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