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

### Example

 If these two bulleted sentences are not immediately obvious, look up assertion and statement and think about them and the examples below for awhile.

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 0.  Even so, the statement  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 MPSetEqnAttrs('eq0028','',3,[[29,13,1,-1,-1],[39,17,1,-1,-1],[50,23,3,-1,-1],[],[],[],[124,54,6,-2,-2]]) MPEquation() MPNNCalcTopLeft(document.mpeq0028ph,'') MPDeleteCode('eq0028')

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 42.)

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 2.

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 4.

¨  Proof by example is a common mistake made in situations like this.

## 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.