1.3 Quantifiers

Put simply, these are just ways of making your life easier, so that you have fewer words to write out when doing maths. There are two quantifiers you’ll come across in common usage:

• Firstly, the phrase ‘for all’ can be represented by \(\forall\).
• Secondly, the phrase ‘there exists’ can be represented by \(\exists\).

For example, consider the statements \[\begin{align*} (\forall x \in \mathbb{R})(x > 2),\\ (\exists x \in \mathbb{R})(x > 2). \end{align*}\] The first one says ‘for all real numbers \(x\), \(x\) is greater than 2’, whereas the second says ‘there exists a real number \(x\) such that \(x\) is greater than 2’. Hopefully you can see that the choice of quantifier can make a huge difference to the truth of a statement! Naturally, for a statement \(P\), these can be negated too in the following ways: \[\begin{align*} \neg(\forall x \; P) &\Leftrightarrow \exists x \; \neg P,\\ \neg(\exists x \; P) &\Leftrightarrow \forall x\; \neg P. \end{align*}\] As a final note, if you have statements in which both quantifiers appear, don’t swap them! For example, suppose \(S \subseteq \mathbb{R}\) is a non-empty set, and consider the two statements \[\begin{align*} (\forall x \in S)(\exists y \in S)(x < y),\\ (\exists y \in S)(\forall x \in S)(x < y). \end{align*}\] The first of these says that \(S\) has no largest element, and is a perfectly valid statement. However, if we choose \(x\) to be equal to \(y\) in the second statement, we can see that this version of the statement is totally impossible!