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Posted 5 June 2009
universally true ASSERTIONS
An assertion containing a variable that is true for any value of the correct type of that variable is called universally true (or a universal assertion).
Example
The assertion is true for any real number
x. In particular,
. You don’t have to calculate out what
and
are to know that they are the same because you
know the assertion P is true for all real numbers.
¨
This assertion is an identity,
which means that it is a universally true equation. (This is one of several
meanings of the word identity). There are many ways in a text to indicate that
an assertion is an identity, for example by saying “For all x, ” or
simply “
is an identity”.
More examples
¨
The statement “ ” is a universally true assertion but it would
not be called an identity.
¨ The statement “ x = x” is a universally true assertion no matter what the type of the variable x is.
¨
The statement “ ” is not a universally true assertion for x
a variable of type real. In fact
it is true only for x =2.
Usage
In some contexts, a universally true assertion is called a law. This word is usually applied to equations, but not always.
Counterexamples
If P(x) is a assertion
and c is some particular value for x for which P(c) is false, then P(x)
is not universally
true. In that case, c is called a counterexample
to the assertion P(x).
Example
is not universally true because
is false.
So
. (Of course, any number less than or equal to
The universal quantifier symbol
The notation x
is used in mathematical logic to denote that the assertion
following it is true of all x.
Example
Suppose x is a real variable. Then the statement “ ” means that for every real number x,
.
It is an example of a universal
statement.
¨
“ ” is an assertion.
It is true for all real x.
¨ “ ” is a statement.
It is true.
More examples
|
|
¨
is false.
Any number is a counterexample.
¨
is true.
¨
is false.
is not true for all x. 0 is a counterexample (it
is the only one).
The symbol is called the
universal quantifier. It is expressed in mathematical English
in a great many ways, some of which cause considerable confusion to people not
in the know. This is discussed below.
Remarks
¨ The usage of the universal quantifier symbol is discussed here.
¨
In a statement such as “For all real x, ” the variable x is said to be universally quantified.
¨
It would be wrong to say that is "almost always true" or
to put any other qualification on it. Any
universal statement is either true or false, period.
¨
The statement " " is true for x = 3 and false for x = 0.
In fact it
is true for every x except
is just plain false.
¨ It
is OK to say “ is almost always true,” because it is true with only one exception. Here I am using “almost always” in its
ordinary sense in conversational English.
In fact, “almost always” has a mathematical definition for which the
statement “
is almost always true” is correct (for real
numbers and some other types of data), although it would be more customary to
say almost surely.
How universal assertions are worded
Here is an incomplete list of the ways the assertions “For
all x, ” might be worded. Universally true conditional assertions
are discussed here.
Symbolic 
The symbolic version can be displayed in any of the following ways:
(all x)
(x)
¨
The first one uses the logical
notation . This is done more often at the
blackboard than in print.
¨ The second and third apparently assume that x is a real variable. Of course, this assumption may be explicit in the full text. Perhaps it occurs in a book entitled “Basic properties of real numbers”!
¨
Using (x) to
mean “for all x” is now old fashioned but you still see it. Note that it is in the form of a constraint
that is not really a constraint -- in contrast to for example “
” thus the statement is true for all x.
¨
The “
” form
(meaning for all x
in 
) is used
in the great classic text Linear Operators by Dunford and
Schwarz. It confused the #*@*%
out of me when I was a graduate student.
More about this here.
All
¨
For all x, 
. This is the way
the symbolic expression “
” is pronounced. This
is a formal way to write the statement because it translates the symbolic
expression symbol by symbol.
¨
A less formal way to write the same expression
would be: “All real numbers x satisfy the inequality .”
Any
¨
For any real number x, .
¨
Or: for any real x.
Every
For every
real number x, .
Each
For each
real number x, . This appears to
me to be uncommon.
Always
¨ is always greater than 0.
¨
Or: 
is always positive.
This use of “always” is noticeably less formal than the other usages above. The image behind it is that you can vary x all over the place as long as you want and the expression stays greater than 0. Of course, in the rigorous view nothing is changing.
Universally true equations
Universally true equations may be given bare:
One clue that the equation is meant
universally is that it might be referred to as an identity or as a law. The symbol “
” may also be used to indicate that it is an identity, as in
but be warned: “” is also used to denote a congruence (in any of that word’s several related meanings). See also constraint.
Distributive plural
A statement in English such as “all squared nonzero real
numbers are positive” is called a distributive plural. This means that the statement “the square of x is positive” is true for every nonzero
real number. It can be translated
directly into symbolic notation: 
.
Not all statements involving plurals in English are distributive plurals. The statement “The agents are surrounding the building” does not imply that Agent James is surrounding the building. This type of statement is called a collective plural. Such a statement cannot be translated directly into a statement involving a universal quantifier. More about this here.
The word “distributive” as used here is analogous to the distributive law of arithmetic. If the set of things referred to is finite,
for example the set {-2, -1, 1, 3} then one can say that “
” is equivalent to “
”.
Exercise
I once found a report on the internet (note) that a Quaker Oats box contained this exhortation: "Eating a good-sized bowl of Quaker Oatmeal for 30 days will actually help remove cholesterol from your body." Think about this in connection with distributive plural, but don’t ask me about it if you get confused, since I don’t understand the linguistic questions it raises myself.
Methods of deduction for universal assertions
Universal instantiation
If 
is a correct mathematical assertion about objects x of some type, and c is some particular object of that type, then it is correct to assert that P(c) is true. This
method of deduction is called universal instantiation.
Example
For real numbers x, the statement 
is correct. Therefore, the statement 
is correct by universal instantiation. (The object c in the preceding paragraph is
You may never see the phrase “universal instantiation” outside of a logic text. This method of deduction is so natural that it is normally used without comment.
Universal generalization
If you have proved P(c)
for an arbitrary object c of some type, and during the proof have
made no restrictions on c, then you are entitled to conclude that P(x)
is true for all x of
the appropriate type. This process is formalized in mathematical logic as the
rule of deduction called universal
generalization. You may have used this method of proof (or seen it used) many times
without having it explicitly stated and named.
Example
Definition:
An integer is even if
it is divisible by
Theorem: Prove that if n is an even integer then so is
.
Proof:
Suppose n is an even integer. By definition, there is an
integer k for which 
. Then 
so is even by definition.
Notes
¨ This proof could begin, “Let n be an even integer” or “Suppose n is an even integer”.
¨ In this proof the only
restriction I made on n was that for some k. I was entitled to do that by
definition of “even”. If I had said, for
example, that. for some k then my
proof would not be correct since not all even
integers are divisible by
¨ Proof by example is a common
mistake made in situations like this.
¨ See also contrapositive method.
Disproving a
universal assertion
You may disprove a universal assertion by finding a counterexample.
Example
“Every odd number is prime.”
Disproof: This statement is false. The number 9 is odd but not prime, so 9 is a counterexample.
Remarks
In using
universal generalization to prove a statement about an arbitrary object c, we are not allowed to make any special
assumptions about c except that it satisfies the hypothesis.
On the other hand, if we suspected that the theorem were false, we could prove
that it is false merely by finding a counterexample:
a single example of c satisfying the
hypothesis
but not the conclusion.
Consider the statement, "Every prime number is odd." The integer 2 is a counterexample. In fact it is the only counterexample,
but the statement "Every prime number is odd" is nevertheless utterly and totally
false. It is not somewhat false, or sometimes
false, it is false.
This phenomenon has been known to give newcomers to abstract math the impression that proving statements is much harder than disproving them, which somehow doesn't seem fair.