Compare for instance how this applies in the case of [multiplication](/Mathematics/Prealgebra/Distributivity.md):
$$
a \cdot (b + c) = a \cdot b + a \cdot c
$$
In addition we have [DeMorgan's Laws](/Logic/Laws_and_theorems.md/DeMorgan's_Laws.md) which express the relationship that obtains between the negations of conjunctive and disjunctive expressions
## Applying the laws to simplify complex Boolean expressions
Say we have the following expression:
$$
\lnot(\lnot(x) \land \lnot (x \lor y))
$$
We can employ DeMorgan's Laws to convert the second conjunct to a different form:
$$
\lnot (x \lor x) = \lnot x \land \lnot y
$$
So now we have:
$$
\lnot(\lnot(x) \land (\lnot x \land \lnot y ))
$$
As we have now have an expression of the form _P and (Q and R)_ we can apply the Distributive Law to simplify the brackets (_P and Q and R_):
$$
\lnot( \lnot(x) \land \lnot(x) \land \lnot(y))
$$
Notice that we are repeating ourselves in this reformulation. We have $\lnot(x) \land \lnot(x)$ but this is just the same $\lnot(x)$ by the principle of **idempotence**. So we can reduce to:
$$
\lnot(\lnot(x) \land \lnot(y))
$$
This gives our expression the form of the first DeMorgan Law ($\lnot (P \land Q)$), thus we can apply the law ($\lnot P \lor \lnot Q$) to get:
$$
\lnot(\lnot(x)) \lor \lnot(\lnot(y))
$$
Of course now we have two double negatives. We can apply the double negation law to:
$$
x \lor y
$$
// TO DO:
- Use truth tables to show equivalence
- Explicitly add implicit laws
- Link to deductive rules
- Link to digital circuits and NANDs as universal gates
