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| | Please come to lecture with the following definitions: | | Please come to lecture with the following definitions: |
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| − | {{:Function definition}} | + | {{Definition:Function}} |
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| − | {{:Surjection definition}} | + | {{Definition:Surjection}} |
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| − | {{:Injection definition}} | + | {{Definition:Injection}} |
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| − | {{:Bijection definition}} | + | {{Definition:Bijection}} |
Revision as of 14:49, 27 January 2020
Last semester's notes
Please come to lecture with the following definitions:
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.
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].
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].
