MATH 302 Lecture 5

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Quantifiers

Definition of a limit

$\forall a\forall \epsilon >0\exists \delta >0\forall x(0<|x-a|<\delta \rightarrow |f(x)-f(a)|<\epsilon )$

Switching $\forall \epsilon$ and $\exists \delta$ 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 $P(x,y)$

$\forall x\forall yP(x,y)$

the entire graph would be filled in

$\exists x\exists yP(x,y)$

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

$\forall x\exists yP(x,y)$

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

$\exists y\forall xP(x,y)$

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

$\exists x\forall yP(x,y)$

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

$\forall y\exists xP(x,y)$

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

Rules of Inference

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