  These laws are better understood in terms of the basic example of a BA, consisting of a collection \(A\) of subsets of a set \(X\) closed under the operations of union, intersection, complementation with respect to \(X\), with members \(\varnothing\) and \(X\). One can easily derive many elementary laws from these axioms, keeping in mind this example for motivation. Any BA has a natural partial order \(\le\) defined upon it by saying that \(x \le y\) if and only if \(x + y = y\).  This corresponds in our main example to \(\subseteq\). Of special importance is the two-element BA, formed by taking the set \(X\) to have just one element. The two-element BA shows the direct connection with elementary logic. The two members, 0 and 1, correspond to falsity and truth respectively. The Boolean operations then express the ordinary truth tables for disjunction (with \(+)\), conjunction (with \(\cdot)\) and negation (with \(-)\). An important elementary result is that an equation holds in all BAs if and only if it holds in the two-element BA. Next, we define \(x \oplus y = (x \cdot -y) + (y \cdot -x)\). Then \(A\) together with \(\oplus\) and \(\cdot\), along with 0 and 1, forms a ring with identity in which every element is idempotent. Conversely, given such a ring, with addition \(\oplus\) and multiplication, define \(x + y = x \oplus y \oplus(x \cdot y)\) and \(-x = 1 \oplus x\). This makes the ring into a BA. These two processes are inverses of one another, and show that the theory of Boolean algebras and of rings with identity in which every element is idempotent are definitionally equivalent. This puts the theory of BAs into a standard object of research in algebra. An atom in a BA is a nonzero element \(a\) such that there is no element \(b\) with \(0 \lt b \lt a\). A BA is atomic if every nonzero element of the BA is above an atom. Finite BAs are atomic, but so are many infinite BAs. Under the partial order \(\le\) above, \(x + y\) is the least upper bound of \(x\) and \(y\), and \(x \cdot y\) is the greatest lower bound of \(x\) and \(y\). We can generalize this: \(\Sigma X\) is the least upper bound of a set \(X\) of elements, and \(\Pi X\) is the greatest lower bound of a set \(X\) of elements. These do not exist for all sets in all Boolean algebras; if they do always exist, the Boolean algebra is said to be complete.