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Posted 15 April 2008

universally true ASSERTIONS

Definition: universally true assertion

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).

Examples

¨  The assertion

    (P)

is true for any real number x.  In particular, 42 is a real number, so we know that .  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).

¨  The statement  “  ” is a universally true assertion but it would not be called an identity.

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 3 is a counterexample to the statement .  (Of course, any number less than or equal to 4 is a counterexample.  Indeed, 4 is a counterexample!)

The universal quantifier symbol

The notation x is used in mathematical logic to denote that the assertion following it is true of all x.

Suppose x is a real variable.

¨   means that for every x, .  It is true.

¨   is false.  Any number is a counterexample.

¨   is true.

¨   is false.  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 0.  Even so, the statement  is just plain false.

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)

Using (x) to mean “for all x” is now old fashioned but you still see it.  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. 

All

¨  For alll 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 .”  But see distributive plural.

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 are often 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.  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 “