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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 ${{x}^{2}}-1=(x-1)(x+1)\,\,\,\,\,\,\,\,\text{(A)}$ is true for any real number $x$.  In particular, $42$ is a real number, so we know that ${{42}^{2}}-1=(42-1)(42+1)$.  You don’t have to calculate out what ${{42}^{2}}-1$  and $(42-1)(42+1)$ are to know that they are the same because you know assertion (A) is true for all real numbers.

Usage

Assertion (A) is an identity, which means that it is a universally true equation. (This is one of several meanings of the word "identity").

In some contexts, a universally true assertion is called a law.  This word is usually applied to equations, but not always.

More examples
• The statement  "$x+3\gt x$" is a universally true assertion for real numbers, but it would not be called an identity because it is not an equation.
• The statement  "$x = x$" is a universally true assertion no matter what the type of the variable $x$ is.
• The statement "$x+3=5$" is not a universally true assertion for $x$ a variable of type real.  In fact it is true only for $x =2$.
• The statement "$x^2\gt0$" is not a universally true assertion for real numbers. It is in fact true for every real number except $x=0$, but that one exception means it is not universally true.

The universal quantifier symbol

The notation "$\forall x$" 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 "$\forall x(x+3\ge x)$" means that for every real number $x$, $x+3\ge x$.  It is an example of a universal statement.

• "$x+3\ge x$" is an assertion. It is true for all real $x$. You can substitute any real number of $x$ and the result is a true statement.
• "$\forall x(x+3\ge x)$" is a statement. It is true. The variable $x$ in this expression is a bound variable. You cannot substitute for it. It would be silly -- more than silly, meaningless -- to write $\forall 6(6+3\ge 6)$".
More examples
• "$\forall x(x\gt x)$" is false.  Any number is a counterexample.
• "$\forall x(x\ge x)$" is true.
• "$\forall x(x^2\gt 0)$" is false. $0$ is a counterexample (it is the only one). It is wrong to say that "$\forall x(x^2\gt0$" is "almost always true" or "just barely false". The statement "$\forall x(x^2\gt0)$" is a claim that "$x^2\gt0$" is true for all real numbers $x$, and that is simply a false statement.

"$\forall x(x^2\gt0)$" is just plain false. Period.

The symbol $\forall$ 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 and briefly in the symbols chapter.

How universal assertions are worded

Here is an incomplete list of the ways the assertion "For all $x$, ${{x}^{2}}+1\gt0$" might be worded. Words in square brackets may be omitted. This list certainly does not cover all possible variations.

Universally true conditional assertions are discussed here.

Symbolic

The symbolic version can be displayed in any of the following ways:

• "$\forall x\,({{x}^{2}}+1\gt 0)$." This form is used more often at the blackboard than in print, except in logic texts.
• ${{x}^{2}}+1\gt 0\,\,\,\,\, (\text{all }x)$.
• ${{x}^{2}}+1\gt 0\,\,\,\,\, (x)$. Using "$(x)$" to mean "for all $x$" is now old fashioned but you still see it. The motivation for this notation is that if you wrote, for example, "$(x\gt1)$" it would put the constraint that $x\gt 1$ on the assertion; so writing "$(x)$" was thought of as an empty constraint -- a constraint that was no constraint. I have never run across any text that explained this notation.
• "$(\text{all } x)$" or "$(x)$" may be replaced by a type expression such as "$(\text{all }x\in \mathbb{R})$" or "$(x\in \mathbb{R})$", which still means "for all $x$ in $\mathbb{R}$".

The "$(x)$" form to mean "for all $x$" is used in the great classic text Linear Operators by Dunford and Schwarz. It confused the hell out of me when I was a graduate student.

All, Any, Every, (Each?)

In the following expressions, "all" can be replaced by "any" or "every". Of course, "$x^2+1\gt0$" can be replaced by "$x^2+1$ is positive".

"For each real number..." doesn't sound wrong to me but I don't recall every seeing it used in print.

• "For all [real] $x$, $x^2+1\gt0$" or "$x^2+1\gt0$ for all [real] $x$".
• "All real numbers $x$ satisfy [the inequality] $x^2+1\gt0$".

"Any" may also be used with a connotation of arbitrary.

Always

• "$x^2+1$ 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: $x^2-1=(x-1)(x+1)$

One clue that the equation is meant universally is that it might be referred to as an identity or as a law.

The symbol "$\equiv$" may also be used to indicate that it is an identity, as in $x^2-1\equiv (x-1)(x+1)$ but be warned: "$\equiv$" is also used to denote a congruence (in any of that word’s several related meanings).  See also constraint.

Methods of deduction for universal assertions

Universal instantiation

If "$\forall x P(x)$" 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 $\forall x\left[x^2-1=(x-1)(x+1)\right]$ is correct.  Therefore, the statement $42^2-1=(42-1)(42+1)$ is correct by universal instantiation. (The object $c$ in the preceding paragraph is $42$.) In other words, the fact that $42^2-1=41\times43$ follows from universal instantiation.

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 that 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 $\forall n\left(\text{If }n\text{ is even, then }n^2\text{ is even}\right)$

Proof: Suppose $n$ is an even integer. By definition of "even" there is an integer $k$ for which $n=2k$. Then $n^2=4k^2=2\cdot2k^2$ so $n^2$ is even by definition.

• This proof uses the method of rewrite according to definitions.
• In this proof, the only restriction I made on $n$ was that $n=2k$ for some integer $k$, which I was entitled to do by definition of "even". In other words, there is no constraint on $n$ other than what is given.
• Proof by example is a common mistake made in proofs like this. ("$6$ is even, and $36$ is even.")

Disproving a universal assertion

You may disprove a universal assertion by finding a counterexample, a value $c$ that makes the assertion false.

Example

"Every odd number is prime."

Disproof: This statement is false.  The number $9$ is odd but not prime, so $9$ is a counterexample.

Remark

To prove $\forall xP(x)$ you have to show that $P(c)$ is true for every single $c$. To prove it is false you need only show that $P(c)$ is false for one value $c$. (*Sigh*) Life is unfair.

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