RELATION OF DIGITAL GATE LOGIC TO CONTACT/COIL DECRIPTION

Now, the lamp will come on if either contact A or contact B is actuated, because all it takes for the lamp to be energized is to have at least one path for current from wire L₁ to wire 1. What we have is a simple OR logic function, implemented with nothing more than contacts and a lamp.
We can mimic the AND logic function by wiring the two contacts in series instead of parallel:
Now, the lamp energizes only if contact A and contact B are simultaneously actuated. A path exists for current from wire L₁ to the lamp (wire 2) if and only if both switch contacts are closed.
The logical inversion, or NOT, function can be performed on a contact input simply by using a normally-closed contact instead of a normally-open contact:
Now, the lamp energizes if the contact is not actuated, and de-energizes when the contact is actuated.
If we take our OR function and invert each "input" through the use of normally-closed contacts, we will end up with a NAND function. In a special branch of mathematics known as Boolean algebra, this effect of gate function identity changing with the inversion of input signals is described by DeMorgan's Theorem, a subject to be explored in more detail in a later chapter. The lamp will be energized if either contact is unactuated. It will go out only if both contacts are actuated simultaneously.
Likewise, if we take our AND function and invert each "input" through the use of normally-closed contacts, we will end up with a NOR function:

A pattern quickly reveals itself when ladder circuits are compared with their logic gate counterparts:

• Parallel contacts are equivalent to an OR gate.
• Series contacts are equivalent to an AND gate.
• Normally-closed contacts are equivalent to a NOT gate (inverter).

We can build combinational logic functions by grouping contacts in series-parallel arrangements, as well. In the following example, we have an Exclusive-OR function built from a combination of AND, OR, and inverter (NOT) gates: