3. Three Important Rules
3.1) Substitution Rule
The Substitution Rule is an important rule in Boolean algebra. It allows for the substitution of equivalent subexpressions for variables or constants in a logical expression without changing the result of the expression.
According to the Substitution Rule, if there is a logical expression E that contains variables or constants, and these variables or constants can be replaced by equivalent logical expressions F, then substituting F for these variables or constants in E will not change the overall result of the expression.
The application of the Substitution Rule can simplify logical expressions, prove equivalences, and facilitate logical reasoning. By selecting appropriate equivalent subexpressions for substitution, complex logical expressions can be simplified in Boolean algebra, making them easier to understand and manipulate.
The Substitution Rule states that for any logical equation, if we replace a variable appearing on both sides of the equation with the same logical function, the equation remains true. This rule is known as the Substitution Rule.
Since logical functions, like logical variables, have only two possible values (0 and 1), the correctness of the Substitution Rule is straightforward to understand. By applying the Substitution Rule, we can extend the application scope of basic laws in logic algebra.
IF F (M(X,Y,Z), B, C…) = G (M(X,Y,Z), B, C …)
F, G, M are functions
IF A = M(X,Y,Z)
F(A,B,C … ) = G(A,B,C …)
is correct
3.1.1 ) Example
A + B = A·B (The Inversion Law)
If we use F = B + C to substitute B in above equation
we can get following equation:
A+B+C=A+F=A·B+C=A·B·C
3.2) Inversion Rule
For any logical function F, if we replace all the operators “·” with “+”, “+” with “·”, constant “0” with “1”, “1” with “0”, original variables with their complements, and complemented variables with their originals, the resulting expression is called the complement or inverse function of the original function F.
The inversion rule is an extension of the De Morgan’s theorem, and it can be used to conveniently derive the complement or inverse function of a given function.
For example:
IF F=AB+C·D+AC
Then:
IF
Then
When applying the inversion rule, two points should be noted:
- The order of operations in the original expression should not be altered. Evaluate the operations within parentheses first, and then follow the “AND before OR” principle.
- The negation (NOT) operator that does not apply to a single variable should be left unchanged.
3.3) Dual Rule
The duality rule in logic algebra states that for any logical expression, if we interchange the AND and OR operators, replace 0 with 1 and 1 with 0, and negate all variables, the resulting expression is the dual of the original expression. The dual rule is a fundamental property that allows us to obtain the dual expression of a given logic function.
For any logic function F, if we replace all the AND operators (·) with OR operators (+), replace all the OR operators (+) with AND operators (·), replace all the constants 0 with 1, replace all the constants 1 with 0, and keep the variables unchanged, we obtain the dual expression of F, denoted as F’ (or F*).
The dual expression represents the logical complement of the original function F.
In each of the examples mentioned above, F’ is the dual expression of F. It can be easily proven that F is also the dual expression of F’. In other words, F and F’ are dual expressions of each other.
Every logical function has a dual expression. If the original equation holds true, then the dual equation will also hold true. This logical reasoning is called the principle of duality or the duality rule. It is important to note that when deriving the dual expression from the original expression, the order of operations should not be changed, and the negation symbols should remain unchanged.
By observing the earlier basic laws and formulas of logic algebra, it is evident that they all appear in pairs and are dual expressions of each other. For example, if the distributive law of multiplication over addition holds true, i.e., A(B+C) = AB+AC, then according to the duality rule, A+BC = (A+B)(A+C), which means the distributive law of addition over multiplication also holds true.