Logic exercises 
1) Make: 
a) a decision table 
b) a decision tree 
for the following problem: 
If you earn less than $10,000 and have at least 2 dependants, you pay
no tax.
If you earn less than $10,000 and have at most 1 dependant, you pay
10%
If you earn at least $10,000 and have at least 3 dependants, you pay
no tax.
If you earn at least $10,000 and have two dependent or fewer, you pay
12%.
2) Show that 
~(A È B) and 
~A É ~B 
are equivalent by means of 
a) a Venn diagram 
b) a truth table 
Two logical expressions, X and Y, in the logical variables A and B are
equivalent if they have the same values for each combination of truth
values for A and B. That is, X has the same truth value as Y when A
and B are both true, when A is true and B is false, etc.

Hello Mohd_fm,

An interesting pair of questions but somewhat hard to illustrate the
answer in text. The ascii art below should illustrate the concepts
required.

1a Decision table

                  You Earn
Dependents   < 10,000  >= 10,000
<2           10% tax   12% tax
2             no tax   12% tax
>2            no tax    no tax

1b Decision tree

              You Earn
             /        \
          < 10,000   >= 10,000
        /      |       |     \
     0-1dep  2+dep   0-2dep  3+dep  
    10% tax  no tax 12% tax  no tax

2 I am going to assume the equations read as...
  not (A or B)
  (not A) and (not B)
are equivalent.... The text on my screen shows the not sign (~), but
not the binary operations (I see a box and accented E). The other
possible choice was
  not (A and B)
  (not A) or (not B)
and I provided the answer for that too. If both are wrong, please
spell out the operator names in a clarification request.

2a. A Venn Diagram

(You should use circles, but I'll approximate with ranges)

|----------------------| (full space)
  | <-- A --> |          (A)
        | <-- B --> |    (B)
  |---------------- |    (A or B)
|-|                 |--| not (A or B) [1]
        |-----|          (A and B)
|-------|     |--------| not (A and B) [2]
|-|           |--------| (not A)
|-------|           |--| (not B)
|-|                 |--| (not A) and (not B) [1]
|-------|     |--------| (not A) or (not B) [2]

Note that the two lines marked with [1] are equivalent, the two lines
marked with [2] are equivalent.

2b A truth table
You can build up the answers below in a way similar to shown for the
Venn Diagrams. I added (A or B) as an example of that.

(A or B)
    A  ~A
 B  1   1
~B  1   0

not (A or B)
    A  ~A
 B  0   0
~B  0   1

(not A) and (not B)
    A  ~A
 B  0   0
~B  0   1

not (A and B)
    A  ~A
 B  0   1
~B  1   1

(not A) or (not B)
    A  ~A
 B  0   1
~B  1   1

There are a number of on line resources for logic problem such as
these. Search with phrases such as
  logic equivalence +and +or +not
to get sites such as
  http://www.chass.utoronto.ca/~osborne/MathTutorial/LOG.HTM

  --Maniac