s2_2 - 2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional...

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2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1 . A tautology is a proposition that is always true. Example 2.1.1 . p ∨ ¬ p Definition 2.1.2 . A contradiction is a proposition that is always false. Example 2.1.2 . p ∧ ¬ p Definition 2.1.3 . A contingency is a proposition that is neither a tautology nor a contradiction. Example 2.1.3 . p q → ¬ r Discussion One of the important techniques used in proving theorems is to replace, or sub- stitute, one proposition by another one that is equivalent to it. In this section we will list some of the basic propositional equivalences and show how they can be used to prove other equivalences. Let us look at the classic example of a tautology, p ∨ ¬ p . The truth table p ¬ p p ∨ ¬ p T F T F T T shows that p ∨ ¬ p is true no matter the truth value of p . [ Side Note. This tautology, called the law of excluded middle , is a direct consequence of our basic assumption that a proposition is a statement that is either true or false. Thus, the logic we will discuss here, so-called Aristotelian logic, might be described as a “2-valued” logic, and it is the logical basis for most of the theory of modern mathematics, at least as it has developed in western culture. There is, however, a consistent logical system, known as constructivist, or intuitionistic, logic which does not assume the law of excluded middle. This results in a 3-valued logic in which one allows for
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2. PROPOSITIONAL EQUIVALENCES 34 a third possibility, namely, “other.” In this system proving that a statement is “not true” is not the same as proving that it is “false,” so that indirect proofs, which we shall soon discuss, would not be valid. If you are tempted to dismiss this concept, you should be aware that there are those who believe that in many ways this type of logic is much closer to the logic used in computer science than Aristotelian logic. You are encouraged to explore this idea: there is plenty of material to be found in your library or through the worldwide web.] The proposition p ∨ ¬ ( p q ) is also a tautology as the following the truth table illustrates. p q ( p q ) ¬ ( p q ) p ∨ ¬ ( p q ) T T T F T T F F T T F T F T T F F F T T Exercise 2.1.1 . Build a truth table to verify that the proposition ( p q ) ( ¬ p q ) is a contradiction. 2.2. Logically Equivalent. Definition 2.2.1 . Propositions r and s are logically equivalent if the statement r s is a tautology. Notation: If r and s are logically equivalent, we write r s. Discussion A second notation often used to mean statements r and s are logically equivalent is r s . You can determine whether compound propositions r and s are logically equivalent by building a single truth table for both propositions and checking to see that they have exactly the same truth values.
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s2_2 - 2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional...

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