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## Inclusive or

Definition 2.3.2   The connector or is defined as follows: if and are two propositions, the     or is true if, and only if, at least one of the components is true. This connector is called the inclusive or and is denoted by the symbol .

The inclusive or is described in the following table.
 T T T T F T F T T F F F
For example:
• the sentence New York is in America or in Australia'' is true because the first component is true.
• The sentence is Galileo Galilei was an Italian or an astronom'' is true because both components are true.
• Consider the following sentence: If Firenze is in Tuscany or the teacher is fair-haired, then Dany will eat an ice cream to-day''. Definitely, Dany will eat an ice cream to-day, because this sentence is true. The fact that we have no idea about the teacher's hair has no influence, as the first component of the statement is true.

Proposition 2.3.3 (Commutative Law)   For any two propositions and , the statement is logically equivalent to .

Proof. We use a truth table.
 T T T T T F T T F T T T F F F F

Proposition 2.3.4 (Associative Law)   For any three propositions , and , the statement is logically equivalent to .

Proof. We use a truth table.
 T T T T T T T T T F T T T T T F T T T T T T F F F T T T F T T T T T T F T F T T T T F F T T T F T F F F F F F F
The fifth column is identical to the last one, whence the claim.

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root 2002-06-10