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.