We have seen how to use the operator to calculate with booleans. It is time to investigate another operator, the supremum, written as `‘ and pronounced `cup’.

has a *higher binding power* than which means that the expression

is read as

(Notice the greater space around in the first expression).

The first two properties of the supremum are familiar:Associativity:

Symmetry:

The next two are new:

**Idempotence:**

**Distributivity/Factoring:**

Armed with these rules, we can now prove our first property concerning the supremum:

**Distributivity/Factoring:**
*Proof*

Our next rule is

**Fixed point***:*
*Proof*

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