Linear Algebra/Propositions

Linear Algebra
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The point at issue in an argument is the proposition. Mathematicians usually write the point in full before the proof and label it either Theorem for major points, Corollary for points that follow immediately from a prior one, or Lemma for results chiefly used to prove other results.

The statements expressing propositions can be complex, with many subparts. The truth or falsity of the entire proposition depends both on the truth value of the parts, and on the words used to assemble the statement from its parts.

For example, where   is a proposition, "it is not the case that  " is true provided that   is false. Thus, "  is not prime" is true only when   is the product of smaller integers.

We can picture the "not" operation with a Venn diagram.


Where the box encloses all natural numbers, and inside the circle are the primes, the shaded area holds numbers satisfying "not  ".

To prove that a "not  " statement holds, show that   is false.

Consider the statement form "  and  ". For the statement to be true both halves must hold: "  is prime and so is  " is true, while "  is prime and   is not" is false.

Here is the Venn diagram for "  and  ".


To prove "  and  ", prove that each half holds.

A "  or  " is true when either half holds: "  is prime or   is prime" is true, while "  is not prime or   is prime" is false. We take "or" inclusively so that if both halves are true "  is prime or   is not" then the statement as a whole is true. (In everyday speech, sometimes "or" is meant in an exclusive way— "Eat your vegetables or no dessert" does not intend both halves to hold— but we will not use "or" in that way.)

The Venn diagram for "or" includes all of both circles.


To prove "  or  ", show that in all cases at least one half holds (perhaps sometimes one half and sometimes the other, but always at least one).



An "if   then  " statement (sometimes written "  materially implies  " or just "  implies  " or " ") is true unless   is true while   is false. Thus "if   is prime then   is not" is true while "if   is prime then   is also prime" is false. (Contrary to its use in casual speech, in mathematics "if   then  " does not connote that   precedes   or causes  .)

More subtly, in mathematics "if   then  " is true when   is false: "if   is prime then   is prime" and "if   is prime then   is not" are both true statements, sometimes said to be vacuously true. We adopt this convention because we want statements like "if a number is a perfect square then it is not prime" to be true, for instance when the number is   or when the number is  .

The diagram


shows that   holds whenever   does (another phrasing is "  is sufficient to give  "). Notice again that if   does not hold,   may or may not be in force.

There are two main ways to establish an implication. The first way is direct: assume that   is true and, using that assumption, prove  . For instance, to show "if a number is divisible by 5 then twice that number is divisible by 10", assume that the number is   and deduce that  . The second way is indirect: prove the contrapositive statement: "if   is false then   is false" (rephrased, "  can only be false when   is also false"). As an example, to show "if a number is prime then it is not a perfect square", argue that if it were a square   then it could be factored   where   and so wouldn't be prime (of course   or   don't give   but they are nonprime by definition).

Note two things about this statement form.

First, an "if   then  " result can sometimes be improved by weakening   or strengthening  . Thus, "if a number is divisible by   then its square is also divisible by  " could be upgraded either by relaxing its hypothesis: "if a number is divisible by   then its square is divisible by  ", or by tightening its conclusion: "if a number is divisible by   then its square is divisible by  ".

Second, after showing "if   then  ", a good next step is to look into whether there are cases where   holds but   does not. The idea is to better understand the relationship between   and  , with an eye toward strengthening the proposition.



An if-then statement cannot be improved when not only does   imply  , but also   implies  . Some ways to say this are: "  if and only if  ", "  iff  ", "  and   are logically equivalent", "  is necessary and sufficient to give  ", " ". For example, "a number is divisible by a prime if and only if that number squared is divisible by the prime squared".

The picture here shows that   and   hold in exactly the same cases.


Although in simple arguments a chain like "  if and only if  , which holds if and only if   ..." may be practical, typically we show equivalence by showing the "if   then  " and "if   then  " halves separately.

Linear Algebra
 ← Appendix Propositions Quantifiers →