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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists x \exists y P(x,y)}

at least one point on the graph satisfies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x, y)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall x \exists y P(x,y)}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists y \forall x P(x,y)}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists x \forall y P(x,y)}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall y \exists x P(x,y)}

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

Rules of Inference

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}