Aside from representing the functioning of a logic gate with a truth
table & a grammatical (with words) definition, the use of logic
equations can be used not only to represent logic gates & circuits,
but also with the usage of some theorems & equivalences, to reduce the
number of terms involved, simplifying the equation.
In logic equation every boolean variable involved is assigned a letter
or symbol, very similar to the algebraic representation of unknown
numerical values using letters; In fact, this approach to logic is
called boolean algebra due to their similarity (remember that it is
called boolean variables & algebra because of the person who did
extensive work on the subject, George Bool).
Each input variable is usually assigned one of the first letters of the
alphabet (A, B, C, & so on), & the output variables are assigned the
last letters (W, Y, Z, & so on; note that X was specifically left
out, this is because it is used as a "Don't Care" condition in logic
simplification). This assignment of letters is arbitrary, any other
letter or symbol can be used instead, but it is a common way to assign
them & most people working in the area follow this pattern.
The logic operations are either written in uppercase (OR, AND, NOT) or
represented by their logical symbol (V for OR, ^ for AND, ~ or the
variable name overlined for NOT). Parenthesis are used to order the
operations & force precedent evaluation before using in other
operations, similar as in algebra where the operations in deeper levels
of parenthesis are evaluated first.
For example, to represent the AND operation, using A & B as input variables & Z as output, you can write
Z = A AND B
or alternatively
Z = A ^ B
For a more complex circuit where the order is not always clear (similar
to algebra, the evaluation is always left to right since there are no
operations of higher priority like division or multiplication are in
mathematical algebra), the use of parenthesis is encouraged, for
example:
Z = A AND B OR B AND C
could mean very different things depending on how it is interpreted, so the equivalent form
Z = (A AND B) OR (B AND C)
being much more explicit in what gets evaluated first is preferred.
Another way to represent the operations in a logical equation is to
simply use the mathematical operators that closely resemble their
operations (+ for OR, * for AND); the NOT gate is an exception to this,
as well as most compound gate. The only compound gates that have a
symbol associated to them are the XOR gate (a + sign enclosed in a
circle) & the XNOR gate (since it represents a logical equivalence,
the = sign or the three line equivalence sign is used)
Truth Tables schematic with explanation
In order to graphically and orderly present each possible output from a
logic gate or any digital circuit, a truth table is used. These tables
present every possible combination of input states and its corresponding
output.
The first columns represent each of the input variables, and the last one (or few if there's more than one) represent the output of the circuit. For low number of variables (lower than 4 or 5) the number of possible combinations is small enough to be able to represent in a truth table, and all possible input combinations and their corresponding output can be quickly visualized
The first columns represent each of the input variables, and the last one (or few if there's more than one) represent the output of the circuit. For low number of variables (lower than 4 or 5) the number of possible combinations is small enough to be able to represent in a truth table, and all possible input combinations and their corresponding output can be quickly visualized
