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Conjunction

Definition 2.3.5        The connector and is defined as follows: if and are two propositions, the     and is true if, and only if, both components are true. This connector is called the conjunction and is denoted by the symbol .

The connector and is described in the following table.
 T T T T F F F T F F F F
For example:
• The sentence Copernic was an astronom and Kikegaard was a philosoph'' is true because both components are true.
• The sentence Copernic was an astronom and Kant run a grocery shop'' is false because the second component is false.
• we cannot decide whether the sentence Copernic was an astronom and lived close to a grocery shop'' is true or not, because we have no possibility to check the truth value of the second component.

Proposition 2.3.6 (Commutative Law)   For any two propositions and , the statement is logically equivalent to .

Proof.
 T T T T T F F F F T F F F F F F
As the two last colums are identical we are done.

Proposition 2.3.7 (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 F F T F T F T F F F F T F F F F F F F T T T F F F F T F F F F F F F T F F F F 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