Download Predicates and Quantifiers

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.
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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.
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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.
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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.
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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
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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
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We have one more piece to put in place before we begin to do a lot
of work translating mathematical statements.
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