CSCI 2824
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Proofs — Existential statements (CSCI 2824, Spring 2015)

In this series of notes, we are going to

  1. Provide proofs of various propositions

  2. Ask you to fix some of our proofs.

  3. Practice proving theorems by expressing your argument in a succinct and logically consistent form.

In this page, we briefly discuss existential statements, and their common proof strategies.

Existential Statements

We will mention existential statements. They are rarer but interesting.

Simple existential statements assert that a number with some property exists.

(exists n in mathbb{N}) P(n).

To prove an existential statement, we just give an example.

Example

Theorem There exist two prime numbers whose sum is also a prime number.

Proof

2 and 3 serve as examples to our theorem.

A first look at constructive Proofs

Existential statements can be proved in another way without producing an example. Typically this involves a proof by contradiction (we will study these types of proofs soon). Such proofs are called non-constructive proofs.

Theorem There exist two irrational numbers a and b such that a^b is rational.

Proof

We will show that such numbers exist without giving you a concrete example.

Consider the number a = sqrt{2}^{sqrt{2}} and b = sqrt{2}.

a^b = (sqrt{2}^{sqrt{2}})^{sqrt{2}} = sqrt{2}^{(sqrt{2}cdot sqrt{2})} = sqrt{2}^2 = 2

Therefore a^b is rational. We know that b is irrational.

There are two cases:

  1. a is irrational. In that case, we are done since a,b are both irrational and a^b is rational.

  2. a is rational.

    1. In that case, a = sqrt{2}^{sqrt{2}} is rational.

    2. But sqrt{2} is irrational itself.

    3. So in this case, we have c=d = sqrt{2} are irrational and c^d = a is rational by assumption.

As a result, there must exist two numbers a,b such that a^b is rational while a,b themselves are irrational. ( Our argument just has not produced any concrete example to point to. :-) )

QED.

Needless to say we will leave non-constructive proofs to mathematicians and the debate to the philosophers for now. If you are interested, these ideas are usually covered in a philosophy of mathematics or a philosophy of science class.