Note: ‘ ` ‘ is used to represent not operation.
a.
X 
Y 
Z 
E 
Min tem of E 
Max Term of E 
F 
Min term of F 
Max term of F 
0 
0 
0 
0 

X`+Y`+Z` 
1 
X`Y`Z` 

0 
0 
1 
1 
X`Y`Z 

0 

X+Y+Z` 
0 
1 
0 
1 
X`YZ` 

1 
X`YZ` 

0 
1 
1 
0 

X+Y`+Z` 
0 

X+Y`+Z` 
1 
0 
0 
1 
XY`Z` 

1 
XY`Z` 

1 
0 
1 
0 

X`+Y+Z` 
0 

X`+Y+Z` 
1 
1 
0 
1 
XYZ` 

0 

X`+Y`+Z 
1 
1 
1 
0 

X`+Y`+Z` 
1 
XYZ 

Note: Min term is written when the function gives ‘1’ as output. While, max term is written, only when the function gives ‘0’ as output.
b.
X 
Y 
Z 
E 
E` 
Min tem of E` 
F 
F` 
Min term of F` 
0 
0 
0 
0 
1 
X`Y`Z` 
1 
0 

0 
0 
1 
1 
0 

0 
1 
X`Y`Z 
0 
1 
0 
1 
0 

1 
0 

0 
1 
1 
0 
1 
X`YZ 
0 
1 
X`YZ 
1 
0 
0 
1 
0 

1 
0 

1 
0 
1 
0 
1 
XY`Z 
0 
1 
XY`Z 
1 
1 
0 
1 
0 

0 
1 
XYZ` 
1 
1 
1 
0 
1 
XYZ 
1 
0 

Note: E` is complement of E i.e. if E = 0 then E`= 1 and if E =1 then E`= 0.
c.
X 
Y 
Z 
E 
F 
E+F 
Min tem of E+F 
E.F 
Min term of E.F 
0 
0 
0 
0 
1 
1 
X`Y`Z` 
0 

0 
0 
1 
1 
0 
1 
X`Y`Z 
0 

0 
1 
0 
1 
1 
1 
X`YZ` 
1 
X`YZ` 
0 
1 
1 
0 
0 
0 

0 

1 
0 
0 
1 
1 
1 
XY`Z` 
1 
XY`Z` 
1 
0 
1 
0 
0 
0 

0 

1 
1 
0 
1 
0 
1 
XYZ` 
0 

1 
1 
1 
0 
1 
1 
XYZ 
0 

Note: ‘+’ is OR operation it gives ‘1’ as output if any of the two input is 1. ‘.’ is AND operation it gives ‘1’ as output if both the input is ‘1’ else it gives ‘0’ as output.
d.
Sum of min terms form of E (using the result from part a.)
= X`Y`Z+X`YZ`+XY`Z`+XYZ`
Note: all the min terms of E which are calculated in part one are added and written.
Sum of min terms form of F (using the result from part a.)
= X`Y`Z`+X`YZ`+XY`Z`+XYZ
Note: all the min terms of F which are calculated in part one are added and written.
e.
Simplifying the result of sum of min term of E –
= X`Y`Z+ X`YZ`+XY`Z`+XYZ` using the min term of ‘E’ from part d.
= X`Y`Z+ X`YZ`+ (XY`Z`+XYZ`)
= X`Y`Z+ X`YZ`+ (Y`+Y)XZ` rearranging and taking XZ` common
= X`Y`Z+ X`YZ`+ 1.XZ` using complementary law
= X`Y`Z+ (X`YZ`+ XZ`) using property of 1: 1.X=X
= X`Y`Z+ (X`Y+ X)Z` rearranging terms and taking Z` common
= X`Y`Z+ (Y+ X)Z` using X+X`Y = X+Y
= X`Y`Z+ YZ`+ XZ`
Hence this is the most simplified form of the sum of min term of E.
Simplifying the result of sum of min term of F –
= X`Y`Z`+X`YZ`+XY`Z`+XYZ using the min term of ‘F’ from part d.
= (X`Y`Z`+X`YZ`)+XY`Z`+XYZ
= (Y`+Y)X`Z`+XY`Z`+XYZ rearranging and taking X`Z` common
= 1.X`Z`+XY`Z`+XYZ using complementary law
= (X`Z`+XY`Z`) +XYZ using property of 1: 1.X=X
= (X`+XY`)Z` +XYZ rearranging terms and taking Z` common
= (X`+Y`)Z` +XYZ using X+X`Y = X+Y
= X`Z` +Y`Z` +XYZ
Hence this is the most simplified form of the sum of min term of F.
Note: X+X`Y = X+Y is also called as third distributive law.