# Simplification

During the simplification step, what simplifications are possible? Each logical operator is fully defined by the rules below. We can use these rules as a look-up table to simplify an expression:

## Not

• $$\lnot 0 = 1$$
• $$\lnot 1 = 0$$

## Or

• $$0 \lor 0 = 0$$
• $$0 \lor 1 = 1$$
• $$1 \lor 0 = 1$$
• $$1 \lor 1 = 1$$

## And

• $$0 \land 0 = 0$$
• $$0 \land 1 = 0$$
• $$1 \land 0 = 0$$
• $$1 \land 1 = 1$$

## Implies

• $$0 \supset 0 = 1$$
• $$0 \supset 1 = 1$$
• $$1 \supset 0 = 0$$
• $$1 \supset 1 = 1$$

Example 1. Can $$\lnot 0$$ be simplified?

Yes. There is one matching rule:

• $$\lnot 0 = 1$$
• $$\lnot 1 = 0$$

So we can replace $$\lnot 0$$ by $$1$$.

Example 2. Can $$x \land y = 1$$ be simplified?

Yes. There is one matching rule:

• $$0 \land 0 = 0$$
• $$0 \land 1 = 0$$
• $$1 \land 0 = 0$$
• $$1 \land 1 = 1$$

So we can infer that $$x = 1$$ and $$y = 1$$.

Example 3. Can $$x \lor y = 1$$ be simplified?

No. There are three matching rules:

• $$0 \lor 0 = 0$$
• $$0 \lor 1 = 1$$
• $$1 \lor 0 = 1$$
• $$1 \lor 1 = 1$$

Neither variable can be pinned down to a single value, so we cannot simplify $$x \lor y = 1$$ any further.