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:

THUS - 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:

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

EXAMPLES:

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:

SO:

Given a Truth Table,

  1. 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

  2. Write an OR expression combining each of the row expressions.

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

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Last Updated 2001/02/15
© L.Heath, 2000, updated by J.A.N. Lee.