Operations of Boolean Algebra are Associative

Theorem
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 1.

Then:


 * $\forall a, b, c \in S: \paren {a \wedge b} \wedge c = a \wedge \paren {b \wedge c}$
 * $\forall a, b, c \in S: \paren {a \vee b} \vee c = a \vee \paren {b \vee c}$

That is, both $\vee$ and $\wedge$ are associative operations.

Proof
Let $a, b, c \in S$.

Let:
 * $x = a \wedge \paren {b \wedge c}$
 * $y = \paren {a \wedge b} \wedge c$

Then:

Similarly:

Thus we see we have $a \vee x = a \vee y$.

Next:

Similarly:

Thus we see we have $\neg a \vee x = \neg a \vee y$.

In conclusion, we have:


 * $a \vee x = a \vee y$
 * $\neg a \vee x = \neg a \vee y$

Hence $x = y$ by Cancellation of Join in Boolean Algebra, that is:


 * $\paren {a \wedge b} \wedge c = a \wedge \paren {b \wedge c}$

The result:


 * $\paren {a \vee b} \vee c = a \vee \paren {b \vee c}$

follows from the Duality Principle.