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Predicates and Qualifiers Sept. 13, 2006 Math 362 1

Predicates and Quantifiers In OEE, we use pronouns to “stand in” for specific nouns: • • He is going to 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 depends 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. Sept. 13, 2006 Math 362 2

Predicates and Quantifiers 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: • • whose truth value depends on what x is. Sept. 13, 2006 Math 362 3

Predicates 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 = x 2}, false if f = {(x, y): x = y 2} p is a convex polygon true if p is x - y is a positive number true if x = 5 and y = 3, false if x = 1 and y=4 Sept. 13, 2006 Math 362 , false if p is 4

Predicates • 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. Sept. 13, 2006 Math 362 5

Quantifiers 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: • Saying something about all members of a particular set, and • Saying something about a particular, but unspecified member of a set. Sept. 13, 2006 Math 362 6

Quantifiers However, many time the quantifiers are implied and have to be supplied by the reader. Example: • • A square is a rectangle. This could mean: • • There is a square that is also a rectangle. (A particular but unspecified square. ) All squares are rectangles (Every square!) This is the more likely interpretation. There is a third possible meaning, namely, • 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. Sept. 13, 2006 Math 362 7

Quantifiers 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. 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: Sept. 13, 2006 Math 362 8

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 x is an optimist, x is a pessimist For all x, x is an optimist, or x is a pessimist. Every square is a rectangle x is a square, x is a rectangle For all x, if x is a square, then x is a rectangle. For all real numbers > 1, x is less than x 2 x is a real number, x > 1, x < x 2 For all x, if x is a real number and x > 1, then x < x 2 Sept. 13, 2006 Math 362 9