Pairs of Quantifiers

Paired quantifiers \(\exists,\exists\)

  • There exists \(x\in A\) so that there exists \(y\in B\) so that \(P(x,y)\)

There exists \(x\in\mathbb{N}\) so that there exists \(y\in\mathbb{N}\) so that \(x+y=5\).

\(\forall,\forall\)

  • For all \(x\in A\) and for all \(x\in B\), \(P(x,y)\).

For all \(x\in\mathbb{N}\) and for all \(y\in\mathbb{N}\), \(xy>0\).

For all \(x\in\mathbb{Z}\) and for all \(y\in\mathbb{N}\), \(xy>0\).

\(\forall, \exists\)

  • For all \(x\in A\) there exists \(y\in B\) so that \(P(x,y)\).

For all \(x\in\mathbb{N}\) there exists \(y\in\mathbb{N}\) so that \(2y=x\).

For all \(x\in\mathbb{Z}\) there exists \(y\in\mathbb{Q}\) so that \(2y=x\).

For all \(\epsilon\in\mathbb{R}\) with \(\epsilon>0\), there exists \(\delta\in\mathbb{R}\) with \(\delta>0\) so that \(x^2<\epsilon\) when \(x<\delta\).

\(\exists, \forall\)

  • There exists \(x\in A\) so that for all \(y\in B\) we have \(P(x,y)\).

There exists \(x\in\mathbb{N}\) so that for all \(y\in\mathbb{N}\) we have \(xy>1\).

There exists \(x\in\mathbb{Q}\) so that for all \(y\in\mathbb{Q}\) we have \(xy<y\).