Matematisk Lingvistik VT96:03

Class:   03
Date:    960125
Topic:   Exercise class 1

Exercise class 1

These exercises are supposed to be handed in on Monday January 29.

Because of the limitation of the range of characters in HTML I cannot display every character I need. I will use the following replacements in this exercise:

A /\ B means: is the intersection of A and B
A \/ B means: is union of A and B
P(A) means the powerset of A
A < B means A is a proper subset of B
A =< B means A is a subset of (and possibly equal to) B

1. In the first two exercises we will work with 5 sets:

   A = { 1 , 2 , 3 , 4 },
   B = { 2 , 3 },
   C = { 3 },
   D = { { 3 } , 4 } and
   E = Ø

   Determine which of the following statements are true:

   1. 2 is an element of A
   2. 3 is an element of B
   3. 2 is an element of C
   4. 3 is an element of D
   5. B =< A
   6. C =< D
   7. { C } =< D
   8. E < E

2. Use the same sets as in the previous exercise and assume that the universe is equal to the union of all these sets. Now compute the elements of
   1. P(D)
   2. B /\ D
   3. A \/ D
   4. ( A - B ) \/ D
   5. B' - A
   6. ( A /\ B )' /\ C

3. Use the laws presented in table 1.7 of the course book to prove that for any sets A, B and C:
   1. ( A \/ B) \/ ( C \/ A ) = ( C \/ B) \/ A
   2. ( A' \/ B)' \/ ( A /\ B ) = A

Every element of the first exercise 1 and 2 is worth 0.5 points. The elements of exercise 3 are worth 1.5 points each.