Discrete Mathematics

Erich Prisner

UMUC SG

Spring 2002

Here you can compute the truth values of 16 statements about the crew of a ship,
consisting of four persons: captain, engineer, sailor, and cook.
You should fill out the following 4*4 table with the ages, heights, education, and gender
of your crew. By clicking on the "Check" button you can evaluate the truth values.

You may also write "T" or "F" in the fields at the beginning of some statements.
Then the long text field shows you where the truth values of the statements considered
differs from your choice. You may also click the button "satisfy conditions"---in that
case the computer tries whether one of 100 random crews satisfies all required conditions.

5)
some member of the crew is female
6) each crew member is at least twice as high as old. 7) no female is larger than 190 (cm). 8) each crew member is larger than 192 (cm) | ||

9) none of the crew members has finished high school. | ||

10) at least one crew member is not older than 30, or female, (or both) | ||

11) if some crew member is female, then the captain is older than the sailor. | ||

12) the captain is female or the engineer is better educated than the cook | ||

13) If there are more males than females on the boat, then the captain has a college degree. | ||

14) if the captain is female, then the engineer is better educated than the cook. |

a) Write the negations of all these statements above in plain English.

b) Complete the Javascript program above at the indicated positions (for instance, between the lines "// Start computation for Aussage[3]" and "// End computation for Aussage[3]" for statement # 3) such that the page also shows the truth values of the statements given in blue (no. 3, ...). "Aussage[3]" is a Boolean variable, use also && and || for "and" and "or". Use != for "not equal" and == for "equal", and note that age, education, height, gender of person i are the variables "Age[i], Educ[i], Height[i], Gender[i]". If you are not familiar with programming languages like javascript, talk to me at my office.

c) Check whether the conjunction of statements # 5, 7, and 8 is satisfiable. That is, find out whether it is possible to create a crew where all the statements # 5, 7, 8 are true. If it is possible, show the crew. If it is impossible, show why not.

d) Check whether the conjunction of statements 4, 10, 11, and the negation of statement #1 is satisfiable.

e) Assume that statements # 4, 10, and 11 are true. Which of the statements above (or their negations) must follow then? Why?

f) Assume that statements # 5, 9, 11, and 12 are true. Which of the statements above (or their negations) must follow then? Why?

f) Express statement # 13 as a compound statement, using only the five prime
statements A_{1}:="The captain is female", A_{2}:="The engineer is
female", A_{3}:="the sailor is female", A_{4}:="the cook is female",
and A_{5}:="the captain has a college degree", and their negations.

Go to Project # 2.

Projects differ in certain ways from homework. They questions there are more complex, more difficult, have several steps, and sometimes you have to use notions and tools from several chapters to solve them. But in most cases, the theorems and tools learned do not suffice to solve all parts of the project. Rather you have to discover certain facts by yourself. Learning how to make "discoveries" in the field is an important part of the class---in real life you donīt have always a textbook telling you exactly what to do. Of course you are encouraged to use any other material---books, webpages, and so on.