Example 1:  The Commutativity of the AND-Operator (&)

In this Example, we introduce the following topics:

• Logical propositions
• The NOT-Operator (~)
• The AND-Operator (&)
• The IMPLIES-Operator (=>)
• The Premise Rule
• The Split Rule
• The Join Rule
• The Conclusion Rule
• The Order of Precedence (Order of Operations)

We will use each of them as we write a very simple, five-line proof. Before we begin writing your proof, however, we need to consider a few preliminaries:
 

Logical Propositions

A logical proposition is just a statement that is considered to be either true or false.

In DC Proof, logical propositions are represented by strings of characters consisting of an uppercase letter followed by any number of letters, digits or primes (apostrophes). The following are all valid examples of logical propositions in DC Proof:

P  A1''  Sunny  Hot

The following strings are the only exceptions. They are key words in DC Proof, and may not be used as logical propositions:

ALL    All
EXIST  Exist
SUM    Sum
PROD   Prod

The NOT-Operator

To indicate, for example,  that the logical proposition "Sunny" is true, we simply write

Sunny

To indicate that it is false, we write

~Sunny

where the tilde (~) symbol is the NOT-operator. This statement is read, "not Sunny."
 

The AND-Operator

Two or more logical propositions also may be combined to form a single statement. To indicate, for example, that the logical propostions "Sunny" and "Hot" are both true we simply write

Sunny & Hot

where the "&" symbol is the AND-operator.

To indicate that "Sunny" is true and "Hot" is false, we write

Sunny & ~Hot

Continued