Math 190 - Predicates and Quantifiers Fall 2003 Page XXX

Math 190 - Predicates and Quantifiers Fall 2003 – Page 1 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

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 true if p is , false if p is polygon

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 3 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. Saying something about all members of a particular set, and 2. 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 4

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!) This is the more likely interpretation.

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 5 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 6 Examples:

x is a real number, x > 1, Math 190 - Predicates and Quantifiers Fall 2003 – Page 7 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 8