Better to know some
... than all
Forms and Definitions of Boolean Expressions
Take as an example the truth table of a three-variable function as shown below. Three variables, each of which can take the values 0 or 1, yields eight possible combinations of values for which the function may be true.
The function has a value 1 for the combinations shown, therefore:
This can also be written as: f(A, B, C) = 000 + 010 + 011 + 111
Note that the summation sign indicates that the terms are "OR'ed" together. The function can be further reduced to the form:
f(A, B, C) = (000, 010, 011, 111)
It is self-evident that the binary form of a function can be written directly from the truth table.
(a) the position of the digits must not be changed
(b) the expression must be in standard sum of products form.
It follows from the last expression that the binary form can be replaced by the equivalent decimal form, namely:
f(A, B, C) = (0,2,3,7)
Product of Sums Representation
From the truth table given above the function has the value 0 for the combinations shown, therefore
Writing the inverse of this function:
Applying De Morgan's Theorem we obtain:
Applying the second De Morgan's Theorem we obtain:
The function is expressed in standard product of sums form.
Thus there are two forms of a function, one is a sum of products form (either standard or normal) as given by expression (1), the other a product of sums form (either standard or normal) as given by expression (4). The gate implementation of the two forms is not the same!
Consider the function:
In binary form: f(A, B, C, D) = (0101, 1011, 1100, 0000, 1010, 0111)
In decimal form: f(A, B, C, D) = (5, 11, 12, 0, 10, 7)
A term is a collection of variables, e.g. ABCD.
A constant is a value or quantity which has a fixed meaning. In conventional algebra the constants include all integers and fractions. In Boolean algebra there are only two possible constants, one and zero. These two constants are used to describe true and false, up and down, go and not go etc.
A variable is a quantity which changes by taking on the value of any constant in the algebraic system. At any one time the variable has a particular value of constant. There are only two values of constants in the system- therefore a variable can only be zero or one. Variables are denoted by letters.
A literal is a variable or its complement
Also known as the standard product or canonic product term. This is a term such as , etc., where each variable is used once and once only.
Also known as the standard sum or canonic sum term. This is a term such as , etc., where each variable is used once and once only.
Standard sum of products form
Also known as the minterm canonic form or canonic sum function. A function in the form of the " sum " (OR) of minterms, e.g:
Standard product of sums form
Also known as the maxterm canonic form or canonic product function. A function in the form of the " product " (AND) of maxterms, e.g:
Sum of products
Also known as the normal sum function. A function in the form of the " sum " of normal product terms, e.g:
Product of sums
Also known as the normal product function. A function in the form of the " product " of normal sum terms, e.g:
Normal (general) sum term
A term such as etc.
Normal (general) product term
A term such as etc.
The name "truth table" comes from a similar table used in symbolic logic, in which the truth or falsity of a statement is listed for all possible proposition conditions. The truth table consists of two parts; one part comparising all combinations of values of the variables in a statement (or algebraic expression), the other part containing the values of the statement for each combination. The truth table is useful in that it can be used to verify Boolean identities.
Consider the following map. The function plotted is
Using algebraic simplification, by using T9a of the Boolean Laws (A + Ä = 1). Referring to the map we can encircle the adjacent cells and infer that A and Ä are not required.
If two occupied cells of a Karnaugh are adjacent, horizontally or vertically (but not diagonally) then one variable is redundant. This has resulted by labelling the map as shown
It is an implicant of a function which does not imply any other implicant of the function.
Prime implicant chart
The chart is used to remove redundant prime implicants. A grid is prepared having all the prime implicants listed down the left and all the minterms of the function along the top. Each minterm covered by a given prime implicant is marked in the appropiate postion.