Functions.html (2395B)
1 <?xml version="1.0" encoding="UTF-8"?> 2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> 3 <html><head><link rel="stylesheet" type="text/css" href="sitewide.css"><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/><meta name="exporter-version" content="Evernote Mac 7.0.3 (456341)"/><meta name="keywords" content="sets"/><meta name="altitude" content="-1.53328013420105"/><meta name="author" content="Alex Balgavy"/><meta name="created" content="2018-03-08 16:02:16 +0000"/><meta name="latitude" content="52.3341924041814"/><meta name="longitude" content="4.867576600778637"/><meta name="source" content="desktop.mac"/><meta name="updated" content="2018-03-08 16:31:17 +0000"/><title>Functions</title></head><body><div><span style="font-weight: bold;">Functions</span></div><div>Function f: A ➝ B — binary relation f of type A × B such that every x ∈ A relates to at most one y ∈ B</div><div><ul><li><br/></li><li>domain: all possible input values</li><li>domain of definition: all input values that actually produce defined output</li><li>codomain: what could be the output of a function (like “integers” with f(x)=2x)</li><li>range/image: what actually is the output of a function (like “even numbers” with f(x)=2x)</li></ul><div><br/></div></div><div>injective: if each element x of domain maps to at most one element y of codomain (“one-to-one”)</div><div>surjective: if each element x of domain maps to at least one element y of codomain; the range is the codomain (“onto”)</div><div>total: if the function is defined for all possible input values (domain)</div><div>bijective: if the function is total, injective, and surjective</div><div><br/></div><div><span style="font-weight: bold;">Composition:</span></div><div>(g o f) of f: A ➝ B and g: B ➝ C</div><div><ul><li>D<span style="vertical-align: sub;">gof</span> ⊆ D<span style="vertical-align: sub;">f</span></li><li>R<span style="vertical-align: sub;">gof</span> ⊆ R<span style="vertical-align: sub;">g</span></li></ul><div><br/></div><div><font style="font-size: 14px;"><span style="font-size: 14px; font-weight: bold;">Inverse:</span></font></div></div><div><span style="font-size: 14px;">a function has an inverse only if it’s injective (one-to-one)</span></div><div><br/></div><div><br/></div></body></html>