I need to have an assignment done for discrete math, step by step detailed answers, especially for proofs.

1. We wish to verify the identity     ,A B C C A B     where A, B, and C are
arbitrary sets.
We have
 
   
   
   
   
definition of complement
commutativity of intersection
de Morgan's law
definition of complement
A B C
A B C
C A B
C A B
C A B
 
  
  
  
  
2. We wish to verify the identity     ,A B C C A B     where A, B, and C are
arbitrary sets, using a membership table.
In the table below, T indicates that a specific object belongs to the given set and F
indicates that it does not.
   A B C A A B A B C A B C A B
T T T F F F T F
T T F F F F T F
T F T F F F T F
T F F F F F T F
F T T T F F T F
F T F T F F T F
F F T T T T F T
F F F T T F F F
     
Since all corresponding entries of  A B C  and  C A B  are the same, these sets
are equal.
3. We are given the set  1,2,3,4,5,6,7,8 .U  We wish to complete the following table:
 
set roster representation of set computer representation of set
3,4,5 ?
? 0,1,0,1,1,1,0,0
? ?
? ?
? ?
A
B
A
A B
A B


The computer representation of A is 0,0,1,1,1,0,0,0 since the elements 3, 4, and 5
belong to A and the elements 1, 2, 6, 7, and 8 do not.
The roster representation of B is  2,4,5,6 since the elements 2, 4, 5, and 6 belong to B
and 1, 3, 7, and 8 do not.
The roster representation of A is  1,2,6,7,8 and the computer representation is
1,1,0,0,0,1,1,1 since the elements 1, 2, 6, 7, and 8 belong to A and 3, 4, and 5 do not.
The roster representation of A B is  4,5 and the computer representation is
0,0,0,1,1,0,0,0 since the elements 4 and 5 belong to both A and B, and hence to
,A B while the elements 1, 2, 3, 6, 7, and 8 do not.
Finally, the roster representation of A B is  1,2,4,5,6,7,8 and the computer
representation is 1,1,0,1,1,1,1,1 since the elements 1, 2, 4, 5, 6, 7, and 8 belong to either
A or B, and hence to ,A B while the element 3 does not.
Thus we have the following completed table:
 
 
 
 
 
set roster representation of set computer representation of set
3,4,5 0,0,1,1,1,0,0,0
2,4,5,6 0,1,0,1,1,1,0,0
1,2,6,7,8 1,1,0,0,0,1,1,1
4,5...