Discover Boolean Equations

In some cases we can create a Truth Table based on the known input and output for a function. From this we need to derive the Boolean formula. This method can also be used occasionally to simplify Boolean expressions. Start with the simplest case - where there is only one output value:

EXAMPLE:

A	B	OUTPUT
F	F	F
F	T	F
T	F	T
T	T	F

SOLUTION:

Create a Boolean expression that is True (1) for the given input and False (0) for all other cases.

We know that:

The AND operation produces a True result in only one case - when all its input is True.
The NOT operator converts False (0) into True (1).

THUS -

where the input is True use that Boolean variable "as is",
where the input is False use the NOT of the input variable to produce True,
and then AND the all terms together.

In the above case (line 3):

A = T, B = F

so we derive:

(A AND NOT B)

(A · ~B)

Check:

A	B	~B	A · ~B
F	F	T	F
F	T	F	F
T	F	T	T
T	T	F	F

CONDITION 1:

In fact we can easily see that if we have ANY number of "input" variables, we can write a Boolean expression that will produce T or 1 (True) in exactly one case when:

We write an expression "ANDing" all the variables together, and
Negating each variable when the value associated with that variable is F or 0.

EXAMPLES:

When P = 1, Q = 0, R = 1, S = 1
then the expression (P && ~Q && R && S) = (1 && ~0 && 1 && 1) is True;
When F = 1, G = 1, H = 0, J = 0
then the expression (F && G && ~H && ~J) = (1 && 1 && ~0 && ~0) is True;
When W = 0, X = 0, Y = 0, Z = 0
then the expression (~W && ~X && ~Y && ~Z) = (~0 && ~0 && ~0 && ~0) is True.

CONDITION 2:

If we have a number of variables and want to produce a Boolean expression whenever any one of them is T or 1 (true), then "ORing" them all together will produce T (or 1) if any one of the expressions is true.

EXAMPLES:

When P = 1, Q = 0, R = 1, S = 1
then the expression (P || Q || R || S) = (1 || 0 || 1 || 1) is True;
When F = 1, G = 1, H = 0, J = 0
then the expression (F || G || H || J) = (1 || 1 || 0 || 0) is True;
When W = 0, X = 0, Y = 0, Z = 0
then the expression (W || X || Y || Z) = (0 || 0 || 0 || 0) is False.

SO:

Given a Truth Table,

Write an AND Expression as above for each row that has T or 1 as the expected output (this expression will be true if and only if the conditions of that row are met); and
Write an OR expression combining each of the row expressions.

The resulting expression then represents the complete functionality of the Truth Table.