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Combinational Logic

Many logic circuits have more than one input and only one output. A truth table is tabular listing of all possible inputs for a logic circuit along with the respective outputs – tabular listing of the input/output relationship of a combinational logic circuit.

OR gate
The OR gate is a circuit that has two or more inputs and whose output is equal to the OR of the inputs.
OR truth table:

x y z
0 0 0
0 1 1
1 0 1
1 1 1

AND gate
When A and B are combined using AND multiplication, the result can be expressed as:
X = A.B
In the AND operation the result will be 1only if all the inputs are 1; for all other cases the result is 0.
AND true table

A B X
0 0 0
0 1 0
1 0 0
1 1 1

Using the true tables above we can conclude:
a) 1+A =1 0.A =0 1.A = A
b) A+A = A A.A =A
c) A +A =1 A.A =0

We take the following expression:
Z=A.B + AC
In ordinary algebra we know that we could take out A as a common factor to give:
Z=A(B+C)
This can be done in boolean algebra also. To prove this we fill the following table:
A
B C AB AC AB+AC B+C A(B+C) AC.AB
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 1 0 0 0 0 1 0 0
0 1 1 0 0 0 1 0 0
1 0 0 0 0 0 0 0 0
1 0 1 0 1 1 1 1 1
1 1 0 1 0 1 1 1 1
1 1 1 1 1 1 1 1 1

In other words AB+AC = A(B+C)

Connect the circuit on the fig.2.1.

fig.2.1.

Using this circuit we prove that AB+AC=A(B+C).

Write expressions for the outputs X,Y and Z in the circuits on fig.2.2.

fig2.2
For the fig.2.2a the expression is:
And its truth table is:

A B C X Y Z
0 0 0 1 1 0
0 0 1 1 1 0
0 1 0 1 1 0
0 1 1 1 0 0
1 0 0 1 1 0
1 0 1 1 1 0
1 1 0 0 1 1
1 1 1 0 0 1

For fig. 2.2b the expression is: AB.C =Z
And its truth table is:


A B C X Z
0 0 0 1 0
0 0 1 1 1
0 1 0 1 0
0 1 1 1 1
1 0 0 1 0
1 0 1 1 1
1 1 0 0 0
1 1 1 0 1

Simple rules for negation:
a) negate each variable individually
b) substitute a sum for every product
c) substitute a product for every sum

Example: