MATH 302 Lecture 5

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Quantifiers

Definition of a limit

$\mathrm{\forall }a\mathrm{\forall }ϵ>0\mathrm{\exists }\delta >0\mathrm{\forall }x(0<|x-a|<\delta \to |f(x)-f(a)|<ϵ)$

Switching $\mathrm{\forall }ϵ$ and $\mathrm{\exists }\delta$ filters the satisfying functions from all continuous functions to only constant functions

Ordering and Negations

Think of the domains of $x$ and $y$ as the axes on a graph, and the points for a chosen (x, y) pair are inputs to the predicate $P(x,y)$

$\mathrm{\forall }x\mathrm{\forall }yP(x,y)$

the entire graph would be filled in

$\mathrm{\exists }x\mathrm{\exists }yP(x,y)$

at least one point on the graph satisfies $P(x,y)$

$\mathrm{\forall }x\mathrm{\exists }yP(x,y)$

"for every vertical line x = x*, there is some point (x*, y) that satisfies P(x*, y)"

$\mathrm{\exists }y\mathrm{\forall }xP(x,y)$

"for some horizontal line y = y*, all points on the line satisfy P(x, y*)"

$\mathrm{\exists }x\mathrm{\forall }yP(x,y)$

"for some vertical line x = x*, all points on the line satisfy P(x*, y)"

$\mathrm{\forall }y\mathrm{\exists }xP(x,y)$

"for every horizontal line y = y*, there is some point (x, y*) that satisfies P(x, y*)"

Rules of Inference

$\begin{array}{l}\mathrm{\forall }x(H(x)\to C(x))\\ \mathrm{\forall }x(\mathrm{\neg }L(x)\vee \mathrm{\neg }S(x))\\ \mathrm{\forall }x(\mathrm{\neg }S(x)\to \mathrm{\neg }C(x))\\ \therefore \mathrm{\forall }x(H(x)\to \mathrm{\neg }L(x))\end{array}$