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Section1.5Proofs Involving Quantifiers

Recall that many of the statements we proved before weren't exactly propositions because they had a variable, like \(x\text{.}\) See Proposition 1.4.4 for an example. We mentioned the strangeness at the time, but now we will confront it. In an example like Proposition 1.4.4, we see that it really is a proposition because it should be interpreted as a statement with a universal quantifier. So, that means we need to figure out what a proof of such a statement looks like.

Let's think about this. If I want to show a statement is true for every positive integer, it seems hopeless because I would need to prove infinitely many statements, one for each positive integer. Since I cannot ever prove infinitely many statements, all is lost. Well, not quite. What if I came up with a proof of my statement for the number one, but then I realized that my proof would work just as well for two? or three? or for any positive integer? In that case, I could work through my proof working with any positive integer, and then when I get to the end, I've proved it for any (i.e., all) positive integer.

Proof

Let \(x\) be an arbitrary object of the universe of discourse.

…

Hence \(P(x)\) is true.

Since \(x\) is arbitrary, \((\forall x) P(x)\) is true.

Now, having dealt with universal quantifiers, we also need to see how to deal with existential quantifiers. The most natural way to prove an existential statement \((\exists x) P(x)\) is to produce a specific \(a\) and show that \(P(a)\) is true for your choice. This requires that we somehow figure out which \(x\) will work.

Proof

Pick \(a\) to be a specific object in the universe of discourse.

…

Hence \(P(a)\) is true.

Therefore, \((\exists x) P(x)\) is true.

Of course, proof techniques can be mixed and matched at will, at least as long as they fit the required form. For example, you could prove an existential statement by contradiction. An interesting thing about a proof of that form is that when you finish, you have proven something exists without actually knowing what it is.

Remark1.5.3

The natural number 1729 is called the the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan.

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

―G. H. Hardy
Proof

Consider \(m = 1729\text{.}\) Note that

\begin{equation*} 1^3 + 12^3 = 1 + 1728 = 1729 = 1000 + 729 = 10^3 + 9^3. \end{equation*}

In the following proof technique, we will show how to prove a unique existential statement. For this, it is important to remember that \((\exists ! x) P(x)\) is just an abbreviation by Remark 1.3.4.

Proof

Prove \((\exists x) P(x).\)

Suppose \(y\) and \(z\) are two elements of the universe for which \(P(y)\) and \(P(z)\) are true.

…

Thus, \(y = z\text{.}\)

Therefore, \((\exists ! x) P(x)\text{.}\)