We were unable to assign a truth value to this statement because we had no information about \(x\text{.}\) It is clear that this sentence is true for some values of \(x\) and false for others:
Just as Result 3.2.7 was a statement about integers, we might choose to study the open sentence over, say, the set \(S = \{0,1,2,3,4\}\text{.}\) In this case we would analyse the truth-values of \(P(x)\) over the domain 41
Again, this terminology is reminiscent of functions.
Notice that the extra bits of text “for some \(x \in S\)” and “for all \(x \in S\)” place restrictions which values of \(x\) we take, and so turn the open sentences into statements. With that extra text we can now assign truth values.
Be careful to punctuate these statements nicely — make sure that it is clear to the reader where the quantifier stops and the open sentence begins. In the case of for-all statements we usually just place a comma:
For there-exists statements we write in “so that” or “such that”, since that is how the statements are typically read. Your busy hard-working mathematician will contract the “so that” to “s.t.”:
It is also generally considered bad style to use \(\exists\) and \(\forall\) in sentences in place of “there exists” and “for all”. Mind you, that doesn’t stop people doing it, but in general, it is okay to do in a mathematical statement or equation, but you should avoid writing them in the middle of paragraphs (except in scratchwork).
Quantifiers are often a point of confusion for students. This can be exacerbated by the number of different ways they can be expressed in written or spoken language. For example, the statement “\(\exists x\in A \st P(x)\)” can be read as
There exists \(x\) in \(A\) so that \(P(x)\) is true.
This is critically important, because it shows us a link between the universal quantifier and the implication. It shows us that:
\begin{equation*}
( \forall x \in A, P(x) ) \equiv (x \in A \implies P(x) )
\end{equation*}
Thankfully it is not too hard to see why — think about their truth values.
\(\forall x \in A, P(x)\) is true provided \(P(x)\) is true for every single \(x\) from \(A\text{.}\) It is false if we can find at least one value of \(x\) from \(A\) so that \(P(x)\) is false.
On the other hand, the implication \(x\in A \implies P(x)\text{,}\) is false when the hypothesis is true, but the conclusion is false. That is, we can find a value of \(x\in A\) so that \(P(x)\) is false. Otherwise the implication is true.
Of course, we don’t really mean for every single possible value of the variable \(x\) taken from the set of all possible things in this and every other universe. We actually mean
\begin{align*}
\forall x \in A, P(x) &\implies Q(x)
\end{align*}
where the set \(A\) is often inferred by context. So when we are talking about even and odd numbers (as above), we really mean
Typically the context is clear, and so it is is just cumbersome 43
And tedious for the hard-working time-pressed mathematician.
to write “\(\forall x \in A,\dots\)” before our statements. When it is us doing the writing, we can look after our reader and try to make sure the context is clear. To this end, a good general rule is:
If you are worried that the reader might not understand the context or that your statements might be open to misinterpretation, then put in more words and more details.
Let \(P(n)\) be the open sentence “\((7n-6)/3 \) is an integer.” over the domain \(\mathbb{Z}\text{.}\) Explain whether the following statements are true
\begin{align*}
\exists n \in \mathbb{Z} \st\amp P(n) \\
\forall n \in \mathbb{Z}, \amp P(n)
\end{align*}
In order for the second statement to be true, no matter which integer \(n\) we choose, the statement \(P(n)\) is true. However, if we pick \(n=1\) then
\begin{equation*}
P(1): \frac{1}{3} \text{ is an integer}
\end{equation*}
Since we cannot pick whatever integer \(n\) we want, and still have \(P(n)\) true, it follows that the statement is false. To be more precise, it is false because there is some \(n\) so that \(P(n)\) is false. In symbols this is:
\begin{equation*}
\exists n \in\mathbb{Z} \st \neg P(n).
\end{equation*}
Notice that in the case of the second statement in the above exercise, we have shown the statement to be false, by demonstrating that its negation is true. This brings us to negating quantifiers.
