MATH 302 Lecture 5

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Quantifiers

Definition of a limit

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

Switching 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 \epsilon} and 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 \delta} filters the satisfying functions from all continuous functions to only constant functions

Ordering and Negations

Think of the domains of 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 x} and 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 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 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}}