SP20:Lecture 4 prep

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Revision as of 15:07, 27 January 2020 by {{GENDER:Mdg39|[math]'"2}} [/math]'"7
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I plan to talk about proof techniques and to start covering the definitions for functions. Last semester, proof techniques were covered in lecture 4 and lecture 5, while function properties were covered in lecture 5 and lecture 6. We'll see how far we get.

Please come to lecture knowing the following definitions:

Definition: Function
If [math]A [/math] and [math]B [/math] are sets, then a function from [math]A [/math] to [math]B [/math] (written [math]f : A \href{/cs2800/wiki/index.php/%E2%86%92}{→} B [/math]) is an unambiguous rule giving, for every input [math]x \href{/cs2800/wiki/index.php/%E2%88%88}{∈} A [/math], an output [math]f(x) \href{/cs2800/wiki/index.php/%E2%88%88}{∈} B [/math]. [math]A [/math] is called the domain of [math]f [/math]; [math]B [/math] is called the codomain.
Definition: Injective
A function [math]f : A \href{/cs2800/wiki/index.php/%5Cto}{\to} B [/math] is injective if, for all [math]x_1 [/math] and [math]x_2 \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math], whenever [math]f(x_1) = f(x_2) [/math], we have [math]x_1 = x_2 [/math].
Definition: Surjective
A function [math]f : A \href{/cs2800/wiki/index.php/%5Cto}{\to} B [/math] is surjective if for every output [math]y \href{/cs2800/wiki/index.php/%5Cin}{\in} B [/math], there exists an input [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math] such that [math]f(x)=y [/math].
Definition: Bijective
A function [math]f:A\href{/cs2800/wiki/index.php/%5Cto}{\to}B [/math] is bijective if it is both injective and surjective.