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Surjective function
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> <i>f</i> such that, for every element <i>y</i> of the function's <a href="/facts/Codomain/MGc43nwQ">codomain</a>, there exists at least one element <i>x</i> in the function's <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> such that <i>f</i>(<i>x</i>) = <i>y</i>. In other words, for a function <i>f</i> : <i>X</i> → <i>Y</i>, the codomain <i>Y</i> is the <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of the function's domain <i>X</i>. It is not required that <i>x</i> be <a href="/facts/Unique_(mathematics)/qBpdIqel">unique</a>; the function <i>f</i> may map one or more elements of <i>X</i> to the same element of <i>Y</i>. </p><p>The term <i>surjective</i> and the related terms <i><a href="/facts/Injective_function/5ES6tqf5">injective</a></i> and <i><a href="/facts/Bijective_function/j4gTuTmW">bijective</a></i> were introduced by <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Nicolas Bourbaki</a>, a group of mainly <a href="/facts/France/CxIeIKM3">French</a> 20th-century <a href="/facts/Mathematician/eaFSR8Fx">mathematicians</a> who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word <i>sur</i> means <i>over</i> or <i>above</i>, and relates to the fact that the <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of the domain of a surjective function completely covers the function's codomain. </p><p>Any function induces a surjection by <a href="/facts/Restriction_of_a_function/JV59DhKy">restricting</a> its codomain to the image of its domain. Every surjective function has a <a href="/facts/Inverse_function/Tg5nnXlu">right inverse</a> assuming the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, and every function with a right inverse is necessarily a surjection. The <a href="/facts/Function_composition/r5Pvnmth">composition</a> of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection. </p>