Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fall 2003 – Page 1 Math 190 - Predicates and Quantifiers Predicates and Quantifiers In OEE, we use pronouns to “stand in” for specific nouns: $ $ $ $ He is going to the theater. It won’t start. She told me to get lost. They never did make it work. How do we know if these statements are true? The truth of these statements depend on who “he” is, what “it” is, who “she” is, and so forth. In OEE, we usually depend on the context of the text or the conversation to make sense of these. In mathematics, we also have placeholders for particular numbers, sets, or mathematical objects. We know them as variables. In fact, we are used to working with statements such as: $ $ x>7 x2 + 7x + 12 = 0 whose truth value depends on what x is. Math 190 - Predicates and Quantifiers Fall 2003 – Page 2 Fall 2003 – Page 3 Math 190 - Predicates and Quantifiers A predicate is a statement containing variables, whose truth depends on the values of the variables. Predicates become propositions only when the variables are replaced by specific objects. x is an integer true if x = -7, false if x = 2/3 f is a function true if f = {(x,y): y = x2}, false if f = {(x,y): x = y2} p is a convex polygon true if p is , false if p is x - y is a true if x = 5 and y = 3, false if x = 1 and y = 4 positive number In fact, most mathematical language uses predicates, and not propositions. That is because we are usually interested in general cases, and not specific cases. For example, we are interested in proving something is true of all triangles, and not just for a specific triangle. For this reason, the LFM needs a way of talking about all triangles at the same time. This is done by using quantifiers. Math 190 - Predicates and Quantifiers Fall 2003 – Page 4 In OEE, we have some specific pronouns that we use when we want to talk in generalities: $ Someone drank the last of the milk and put the carton back in the refrigerator. $ Everyone who failed the test lined up outside the professor’s office. $ All of my friends are going to the party. $ There has to be someone who can babysit Calvin tonight. Notice that these talk about either 1) everyone in a specific category (those who failed the test, my friends), or 2) a particular but unspecified person (the person who drank the milk, the poor, hapless babysitter). These are two ways of generalizing that are done all the time in mathematics: 1. 2. Saying something about all members of a particular set, and Saying something about a particular, but unspecified member of a set. However, sometimes the quantifiers are implied and have to be supplied by the reader. Math 190 - Predicates and Quantifiers Fall 2003 – Page 5 Example: A square is a rectangle. This could mean: 1. There is a square that is also a rectangle. (A particular but unspecified square.) 2. All squares are rectangles (Every square!) likely interpretation. This is the more There is a third possible meaning, namely, 3. Some squares are rectangles. But for a mathematician, this says pretty much the same thing as #1 above. If there is one square that is a rectangle, that means that “some” squares are rectangles. In summary, there are two kinds of quantifiers that we use with predicates: one that says a particular, unspecified thing exists, and one that says something is true of all objects. Math 190 - Predicates and Quantifiers Fall 2003 – Page 6 The FLM phrases that usually express these ideas are: $ For every x, . . . . . $ There exists an x such that. . . We combine these phrases with predicates containing the variable x, to make general statements: Math 190 - Predicates and Quantifiers Fall 2003 – Page 7 Fall 2003 – Page 8 Math 190 - Predicates and Quantifiers Examples: OEE Predicate(s) FLM Someone ate all my porridge. X ate all my porridge. There exists an x such that x ate all my porridge. Everything is beautiful. x is beautiful. For all x, x is beautiful. Everybody’s gone surfing (Surfing USA . .) x has gone surfing For all x, x has gone surfing. There is a doctor in the house. x is a doctor, x is in the house There is an x such that x is a doctor and x is in the house. There are two kinds of people: optimists and pessimists Every square is a rectangle x is an optimist, x is a pessimist For all x, x is an optimist, or x is a pessimist. x is a square, x is a rectangle For all x, if x is a square, x is a rectangle. For all real numbers > 1, x is x is a real number, x > 1, For all x, if x is a real number less than x2 and x > 1, then x < x2 Math 190 - Predicates and Quantifiers Fall 2003 – Page 9 Math 190 - Predicates and Quantifiers Fall 2003 – Page 10 We have one more piece to put in place before we begin to do a lot of work translating mathematical statements. Math 190 - Predicates and Quantifiers Fall 2003 – Page 11