Truth Table Description of Logic Functions



Truth Table Description of Logic Functions

The truth table is a tabular representation of a logic function.   It gives the value of the function for all possible combinations of values of the variables.   If there are three variables in a given function, there are 23 = 8 combinations of these variables.  For each combination, the function takes either 1 or 0.  These combinations are listed in a table, which constitutes the truth table for the given function.  Consider the expression,

The information contained in the truth table and in the algebraic representation of the function are the same.

The term ‘truth table’ came into usage long before Boolean algebra came to be associated with digital electronics. Boolean functions were originally used to establish truth or falsehood of statements.  When statement is true the symbol "1" is associated with it, and when it is false "0" is associated. This usage got extended to the variables associated with digital circuits.   However, this usage of adjectives "true" and "false" is not appropriate when associated with variables encountered in digital systems.   All variables in digital systems are indicative of actions.  Typical examples of such signals are "CLEAR", "LOAD", "SHIFT", "ENABLE", and "COUNT".  These are suggestive of actions. Therefore, it is appropriate to state that a variable is ASSERTED or NOT ASSERTED than to say that a variable is TRUE or FALSE.   When a variable is asserted, the intended action takes place, and when it is not asserted the intended action does not take place.  In this context we associate "1" with the assertion of a variable, and "0" with the non-assertion of that variable.  It should now be read as "F is asserted when A and B are asserted or A is asserted and B is not asserted".  This convention of using "assertion” and “non-assertion" with the logic variables will be used in all the Learning Units of this course on Digital Systems.

The term ‘truth table’ will continue to be used for historical reasons. But we understand it as an input-output table associated with a logic function, but not as something that is concerned with the establishment of truth.

As the number of variables in a given function increases, the number of entries in the truth table increases exponentially.  For example, a five variable expression would require 32 entries and a six-variable function would require 64 entries.  It, therefore, becomes inconvenient to prepare the truth table if the number of variables increases beyond four.  However, a simple artefact may be adopted.   A truth table can have entries only for those terms for which the value of the function is "1", without loss of any information.   This is particularly effective when the function has only a small number of terms.
Uploaded Thu, 21-Jan-2021
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