Homework SourseWrite the Boolean expression for output X in the figure belo
Solution
X = NOT(NOT(A OR B) AND NOT(C’ OR B))
The first gate is NOR gate and input are A , B so NOT(A OR B)
Second gate is NOR gate and input are NOT C , B so NOT(C\' OR B)
Third gate is NAND gate and inputs are ouputs of first two gates.
We will simply the expression using DeMorgan\'s Theroem and its says break.
1) First break the longest bar(longest NOT which is becuase of NAND gate) so here multiplication becomes addition (AND becomes OR)
NOT(A OR B) OR NOT(NOT C OR B)
Now break another biggest bars
(A AND B) OR (NOT C AND B)
Now Apply identity to all
(NOT A AND NOT B) OR (C AND NOT B)
Truth table
A
B
C
X = NOT (NOT(A OR B) AND NOT(NOT C OR B))
F
F
F
NOT (NOT (F OR F) AND NOT(NOT C OR B) = NOT(NOT(F) AND NOT(T OR F)) = T
F
F
T
NOT(NOT(F OR F) AND NOT(NOT T OR F)) = NOT(NOT F AND NOT F) T
F
T
F
NOT(NOT(F OR T) AND NOT (NOT F OR T)) = NOT(NOT T AND NOT(T)) = T
F
T
T
NOT(NOT(F OR T) AND NOT(NOT T OR T)) = NOT(NOT T AND NOT T) = F
T
F
F
NOT(NOT (T OR F) AND NOT(NOT F OR F)) = NOT (NOT T AND NOT F) = T
T
F
T
NOT(NOT (T OR F) AND NOT(NOT T OR F) = NOT(F AND NOT F) = T
T
T
F
NOT(NOT(T OR T) AND NOT(NOT F OR T)) = NOT (F AND NOT T) = T
T
T
T
NOT(NOT(T OR T) AND NOT(NOT T OR T))= NOT(F AND NOT T) = T
| A | B | C | X = NOT (NOT(A OR B) AND NOT(NOT C OR B)) |
| F | F | F | NOT (NOT (F OR F) AND NOT(NOT C OR B) = NOT(NOT(F) AND NOT(T OR F)) = T |
| F | F | T | NOT(NOT(F OR F) AND NOT(NOT T OR F)) = NOT(NOT F AND NOT F) T |
| F | T | F | NOT(NOT(F OR T) AND NOT (NOT F OR T)) = NOT(NOT T AND NOT(T)) = T |
| F | T | T | NOT(NOT(F OR T) AND NOT(NOT T OR T)) = NOT(NOT T AND NOT T) = F |
| T | F | F | NOT(NOT (T OR F) AND NOT(NOT F OR F)) = NOT (NOT T AND NOT F) = T |
| T | F | T | NOT(NOT (T OR F) AND NOT(NOT T OR F) = NOT(F AND NOT F) = T |
| T | T | F | NOT(NOT(T OR T) AND NOT(NOT F OR T)) = NOT (F AND NOT T) = T |
| T | T | T | NOT(NOT(T OR T) AND NOT(NOT T OR T))= NOT(F AND NOT T) = T |
