# Simplification

What simplifications are valid?
The tables below *completely specify* the behavior of the logical operators.
Because they are complete, we can use them to see if any simplifications are possible:

**Example 1.** Can be simplified?

From the tables, there is only one possibility:

So .

**Example 2.** Can be simplified?

This time, there are three possibilities:

Neither variable can be pinned down to a single value, so we cannot simplify any further.

**Example 3.** Can be simplified?

Let’s start by considering only the outermost layer of the left-hand expression. To make this clearer, let’s define a variable: . Now the problem becomes: Can be simplified?

There is only one possibility:

So .

Now we can eliminate the variable , and the problem becomes: Can be simplified?

There is only one possibility:

So and .

We can apply these simplifications recursively to unravel any knot!