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When And Means Or
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In English and other natural languages, the words and and or are used in a wide variety of situations. In some of these situations they have meanings that seem to contradict their meanings as logical operators. Because of this, you should never be hasty when you attempt to express a real-world notion logically. In the WHERE clause of a query, combining conditions with AND serves to make the number of rows in the result set smaller. However, the English and often corresponds not to the AND of a query s WHERE clause but to the logical operator OR or the set operator UNION. Consider the following English request: Please bring me the latest invoices for customer 45 and customer 17. This doesn t translate into the query predicate custid=45 AND custid=17. Instead, it probably translates into the query predicate custid=45 OR custid=17. On the other hand, this English request doesn t follow the same pattern: Please bring me the latest recipes for ham and eggs.
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In English, when or doesn t mean and, it still doesn t always mean the same thing as logical or. Logical or means one or the other or possibly both. Sometimes the English word means one or the other but not both, which in a mathematical discussion is distinguished by the name exclusive or. An example of this can be found on many restaurant menus in the phrase includes soup or salad.
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Logical Equivalence
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Two value expressions of any kind are considered equal if they have the same value: 3+3 equals 6. Expressions that contain variables are considered equal if they are equal for any particular variable values: Regardless of what x, y, and z happen to be, {x,y,z} = {a,x,y,z} {b,x,y,z}. Predicates, which are logical propositions containing variables, are said to be logically equivalent if they have the same truth value for any particular values of their variables. Several different symbols are used to represent logical equivalence and some very similar notions. I won t get into any of the subtleties, and from among the possible symbols, which include , , and , I ll use the last one, the bidirectional double arrow.
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DeMorgan s Laws
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Logical expressions can be rewritten as equivalent logical expressions in a number of ways. Two of the most useful and important identities provide ways to rewrite negations, and they are called DeMorgan s Laws, after Augustus DeMorgan.
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Statement of DeMorgan s Laws
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Let P and Q be propositions. Then the following equivalences hold: (P Q) (P Q) ( P) ( Q). ( P) ( Q).
Logical Implication
Mathematical logic was developed largely as an attempt to justify the way in which mathematicians prove theorems through inference and deduction. One of the most important rules of inference is called modus ponens. Modus ponens is the rule of inference that allows us to infer the truth of one proposition Q from the truth of another proposition P when it s known that P implies Q. An argument using modus ponens might go like this: The law is clear: if you drive faster than 55 miles per hour on this highway, you have broken the law. You were driving faster than 55 miles per hour, therefore you have broken the law.
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Set Theory and Predicate Logic
Logical inference isn t the focus of this chapter, but we will take a moment to consider propositions that take the form of logical implication.
If P, Then Q
Suppose P and Q are valid logical propositions. Then if P, then Q is a valid logical proposition. The proposition if P, then Q is denoted P Q, and its truth value depends on the truth values of P and Q as follows.
De nition of P Q
The proposition P Q, read P implies Q or if P, then Q, is true when either P is false or Q is true (or both). The proposition P Q is false when P is true and Q is false. More ( P Q). concisely, (P Q)
There is more than one way to express an implication in words, and in mathematical logic, the following expressions are taken to have the precise meanings shown: 1. P unless Q means ( Q) P. 2. P only if Q means P Q. 3. P, if Q means Q P. Note that unlike the logical operators and , the operator values of P Q and Q P are not necessarily the same. is not commutative. The truth
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