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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science February 1, 2005 Srini Devadas and Eric Lehman Lecture Notes Logic It’s really sort of amazing that people manage to communicate in the English language. Here are some typical sentences: 1. “You may have cake or you may have ice cream.” 2. “If pigs can ﬂy, then you can understand the Chernoff 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 desert? If the second sentence is true, then is the Chernoff 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 they each have a different dream? Some uncertainty is tolerable in normal conversation. But when we need to formu- late ideas precisely— as in mathematics— the ambiguities inherent in everyday language become a real problem. We can’t hope to make an exact argument if we’re not sure ex- actly what the individual words mean. (And, not to alarm you, but it is possible that we’ll be making an awful lot of exacting mathematical arguments in the weeks ahead.) So be- fore 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 mathematicians endow these words with definitions more precise than those found in an ordinary dictio- nary. Without knowing these definitions, you could sort of read this language, but you would miss all the subtleties and sometimes have trouble following along. Surprisingly, in the midst of learning the language of logic, we’ll come across the most important open problem in computer science— a problem whose solution could change the world. 2 Logic 1 Propositions A proposition is a statement that is either true or false. This definition is a little vague, but it does exclude sentences such as, “What’s a surjection, again?” and “Learn logarithms!” Here are some examples of propositions. “All Greeks are human.” “All humans are mortal.” “All Greeks are mortal.” Archimedes spent a lot of time fussing with such propositions in the 4th century BC while developing an early form of logic. These are perfectly good examples, but we’ll be more concerned with propositions of a more mathematical ﬂavor: “ 2 + 3 = 5 ” This proposition happens to be true. Sometimes the truth of a proposition is more difficult to determine: 4 “ a + b 4 + c 4 = d 4 has no solution where a , b , c are positive integers.” Euler...
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