# 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!