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MIT6_042JF10_lec01 - "mcs-ftl 2010/9/8 0:40 page 5#11 1...

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1 “mcs-ftl” — 2010/9/8 — 0:40 — page 5 — #11 Propositions Definition. A proposition is a statement that is either true or false. For example, both of the following statements are propositions. The first is true and the second is false. Proposition 1.0.1. 2 + 3 = 5. Proposition 1.0.2. 1 + 1 = 3. Being true or false doesn’t sound like much of a limitation, but it does exclude statements such as, “Wherefore art thou Romeo?” and “Give me an A !”. Unfortunately, it is not always easy to decide if a proposition is true or false, or even what the proposition means. In part, this is because the English language is riddled with ambiguities. For example, consider the following statements: 1. “You may have cake, or you may have ice cream.” 2. “If pigs can fly, then you can understand the Chebyshev bound.” 3. “If you can solve any problem we come up with, then you get an A for the course.” 4. “Every American has a dream.” What precisely do these sentences mean? Can you have both cake and ice cream or must you choose just one dessert? If the second sentence is true, then is the Chebyshev bound incomprehensible? If you can solve some problems we come up with but not all, then do you get an A for the course? And can you still get an A even if you can’t solve any of the problems? Does the last sentence imply that all Americans have the same dream or might some of them have different dreams? Some uncertainty is tolerable in normal conversation. But when we need to formulate ideas precisely—as in mathematics and programming—the ambiguities inherent in everyday language can be a real problem. We can’t hope to make an exact argument if we’re not sure exactly what the statements mean. So before we start into mathematics, we need to investigate the problem of how to talk about mathematics. To get around the ambiguity of English, mathematicians have devised a special mini-language for talking about logical relationships. This language mostly uses ordinary English words and phrases such as “or”, “implies”, and “for all”. But
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6 “mcs-ftl” — 2010/9/8 — 0:40 — page 6 — #12 Chapter 1 Propositions mathematicians endow these words with definitions more precise than those found in an ordinary dictionary. Without knowing these definitions, you might sometimes get the gist of statements in this language, but you would regularly get misled about what they really meant. Surprisingly, in the midst of learning the language of mathematics, we’ll come across the most important open problem in computer science—a problem whose solution could change the world. 1.1 Compound Propositions In English, we can modify, combine, and relate propositions with words such as “not”, “and”, “or”, “implies”, and “if-then”. For example, we can combine three propositions into one like this: If all humans are mortal and all Greeks are human, then all Greeks are mortal.
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