MATH 301 In Class Exercise on Truth in Structures
Let LG = {0,+} be the language of group theory. Let U be the LG-structure defined as follows:
The universe of U is A = {0,1}. 0U is 0 in A. + is defined on A2 by the following:
0 + 0 = 0, 1 + 0 = 1, 0 + 1 = 1, 1 + 1 = 0.
Finally define the variable assignment function, s, by s(vk) = 0 if k is even and 1 otherwise.
1. The term assignment function 
.
Calculate for each of the
following terms:
a) v3 b) 0 c) 
d) 
2. Does U╞
for each of the following formulas? Remember that s has been defined above.
a) 
b) 
c) 
d) 
e) 
Now let U have the
universe A = {0,1,2,3,4,5}.
Let 0U be 0, let s be defined by 
is the remainder of k when divided by 6. (s(v8) = 2). Finally let + be defined by the following
table:
|
+ |
0 |
1 |
2 |
3 |
4 |
5 |
|
0 |
0 |
1 |
2 |
3 |
4 |
5 |
|
1 |
1 |
2 |
0 |
5 |
3 |
4 |
|
2 |
2 |
0 |
1 |
4 |
5 |
3 |
|
3 |
3 |
4 |
5 |
0 |
1 |
2 |
|
4 |
4 |
5 |
3 |
2 |
0 |
1 |
|
5 |
5 |
3 |
4 |
1 |
2 |
0 |
3. Does U╞ 
? The important task here is to understand exactly what you
have to check. By understanding what the formula means you should be able to
answer the question with a small amount of work.
4. Let 
be the formula 
. It is true that U╞. Check it in two cases.