Solutions are only given for interesting problems.
For each of the three equations show below do the following actions:
a | b | c | (a + b' + c')(a + c) |
0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 |
f(a, b, c) = a' b' c + a b' c' + a b' c + a b c' + a b c
f(a, b, c) = Σ(1, 4, 5, 6, 7)
f(a, b, c) = (a + b + c) (a + b' + c) (a + b' + c')
f(a, b, c) = Π(0, 2, 3)
There are many possible answers. This one is derived from the
optimized SOP expression a + a' b' c .
x | y | z | (x(y + z'))' |
0 | 0 | 0 | 1 |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 1 |
0 | 1 | 1 | 1 |
1 | 0 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 0 |
1 | 1 | 1 | 0 |
f(x, y, z) = x' y' z' + x' y' z + x' y z' + x y' z' + x y' z
f(x, y, z) = Σ(0, 1, 2, 3, 5)
f(x, y, z) = (x' + y + z) (x' + y' + z) (x' + y' + z')
f(x, y, z) = Π(4, 6, 7)
This one is derived from the original expression. The
(y + z')
is replaced by its equivalent, under de Morgan's theorem,
(y' z)' .
a | b | c | (a ⊕ b') + a c' |
0 | 0 | 0 | 1 |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 |
1 | 0 | 1 | 0 |
1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 |
f(a, b, c) = a' b' c' + a' b' c + a b' c' + a b c' + a b c
f(a, b, c) = Σ(0, 1, 4, 6, 7)
f(a, b, c) = (a + b' + c) (a + b' + c') (a' + b + c')
f(a, b, c) = Π(2, 3, 5)
Here the exclusive-OR of
(a ⊕ b') + a c'
is expanded to give the SOP expression
a b + a' b'
+ a c' which is then
implemented with NANDs.
Note: ⊕ is the "exclusive-OR" function.
Do the following for the circuit drawn below:
f(X, Y, Z) = (X' + X Y + X' Y' Z')' (X' Y' Z' + Z)
f(X, Y, Z) = (X' + X Y + X' Y' Z')' (X' Y' Z' + Z) | |
f(X, Y, Z) = (X' + X' Y' Z' + X Y)' (X' Y' Z' + Z) | Associativity |
f(X, Y, Z) = (X' + X Y)' (X' Y' Z' + Z) | Absorption |
f(X, Y, Z) = (X' + Y)' (X' Y' Z' + Z) | Absorption (form 2) |
f(X, Y, Z) = X Y' (X' Y' Z' + Z) | de Morgan's |
f(X, Y, Z) = X Y' X' Y' Z' + X Y' Z | Distributivity |
f(X, Y, Z) = 0 + X Y' Z | Annihilation, Associativity, and Complementarity |
f(X, Y, Z) = X Y' Z | Identify |
One AND gate
f'(X, Y, Z) = (X Y' Z)'
f'(X, Y, Z) = X' + Y + Z'
A control panel for a rudimentary robot has three Control switches and one Power switch. Each switch has two positions -- ON or OFF. The control switches are labeled x , y, and z. The Power switch is labeled p. In order for the robot to be Enabled), the Power switch must be ON, and exactly two of the Control switches must ON. Design the logic circuit to generate the Enable signal. Show your results with a truth table, Boolean (algebraic) expression, and the logic circuit diagram.
Simplify the following expressions using only boolean algebra. Show all your steps for credit.
F(x, y, z) = x + x' y + z | |
F(x, y, z) = x + y + z | Absorption (2nd form) |
F(x, y, z) = x + y + x z + (x y z)' | |
F(x, y, z) = x + y + x z + x' + y' + z' | de Morgan's |
F(x, y, z) = x + x' + y + x z + y' + z' | Associativity |
F(x, y, z) = 1 | Complementarity and Annihilation |
F(x, y, z) = (x + y)(x' + z)(y + z') | |
F(x, y, z) = (x x' + x z + y x' + y z)(y + z') | Distributivity |
F(x, y, z) = (x z + y x' + y z)(y + z') | Complementarity and Identity |
F(x, y, z) = x z y + y x' y + y z y + x z z' + y x' z' + y z z' | Distributivity |
F(x, y, z) = x y z + x' y + y z + 0 + x' y z' + 0 | Annihilation, Associativity, Complementarity, and Idempotency |
F(x, y, z) = x y z + x' y + y z + x' y z' | Identity |
F(x, y, z) = (x y z + y z) + (x' y + x' y z') | Associativity |
F(x, y, z) = y z + x' y | Absorption |
F(x, y, z) = ((((x + y)' + z)' + x)' + y')' | |
F(x, y, z) = (((x + y)' + z)' + x)'' y'' | de Morgan's |
F(x, y, z) = (((x + y)' + z)' + x) y | Involution |
F(x, y, z) = (((x + y)'' z') + x) y | de Morgans' |
F(x, y, z) = ((x + y) z' + x) y | Involution |
F(x, y, z) = (x z' + y z' + x) y | Distributivity |
F(x, y, z) = (x + x z' + y z') y | Associativity |
F(x, y, z) = (x + y z') y | Absorption |
F(x, y, z) = x y + y z' y | Distributivity |
F(x, y, z) = x y + y z' | Associativity and Identity |
F(x, y, z) = x (y + z(x + y z') + x' z') x' | |
F(x, y, z) = x x' (y + z(x + y z') + x' z') | Associativity |
F(x, y, z) = 0 (y + z(x + y z') + x' z') | Complementarity |
F(x, y, z) = 0 | Annihilation |
F(x, y, z) = x + x' z + x' z' | |
F(x, y, z) = x + x' (z + z') | Associativity |
F(x, y, z) = x + x' 1 | Complementarity |
F(x, y, z) = x + x' | Identity |
F(x, y, z) = 0 | Complementarity |