Aside from representing the functioning of a logic gate with a truth table and a grammatical (with words) definition, the use of logic equations can be used not only to represent logic gates and circuits, but also with the usage of some theorems and 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 and 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, and so on), and the output variables are assigned the last letters (W, Y, Z, and 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 and 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 and 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 and B as input variables and 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) and the XNOR gate (since it represents a logical equivalence, the = sign or the three line equivalence sign is used).
