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<h1 align="center">Injective, Surjective and Bijective</h1>
<p align="center">"Injective, Surjective and Bijective" tells us about how a function behaves.</p>
<p>A <a href="function.html">function</a> is a way of matching the members of a set "A" <b>to</b> a set "B":
</p><br>
<div align="center">
<p><img src="images/function-mapping.svg" alt="General, Injective, Surjective and Bijective Functions"></p>
</div>
<p>&nbsp;</p>
<p>Let's look at that more closely:</p>
<div class="dotpoint">
<p>A <b>General Function</b> points from each member of "A" to a member of "B".</p>
<p>It <b>never</b> has one "A" pointing to more than one "B", so <b>one-to-many is not OK</b> in a function (so something like "f(x) = 7 <i><b>or</b></i> 9" is not allowed)</p>
<p>But more than one "A" can point to the same "B" (<b>many-to-one is OK</b>)</p>
</div>
<div class="dotpoint">
<p><b>Injective</b> means we won't have two or more "A"s pointing to the same "B". </p>
<p>So <b>many-to-one is NOT OK</b> (which is OK for a general function).</p>
<p>As it is also a function<b> one-to-many is not OK</b> </p>
<p>But we can have a "B" without a matching "A"</p>
<p>Injective is also called "<b>One-to-One</b>"</p>
</div>
<div class="dotpoint">
<p><b>Surjective</b> means that every "B" has <b>at least one</b> matching "A" (maybe more than one).</p>
<p>There won't be a "B" left out. </p>
</div>
<div class="dotpoint">
<p><b>Bijective</b> means both Injective and Surjective together. </p>
<p>Think of it as a "perfect pairing" between the sets: every one has a partner and no one is left out.</p>
<p>So there is a perfect "<b>one-to-one correspondence</b>" between the members of the sets. </p>
<p>(But don't get that confused with the term "One-to-One" used to mean injective).</p>
<div class="center80">
<p>Bijective functions have an <b>inverse</b>! </p>
<p>If every "A" goes to a unique "B", and every "B" has a matching "A" then we can go back and forwards without being led astray.</p>
<p>Read <a href="function-inverse.html">Inverse Functions</a> for more.</p>
</div>
</div>
<h2>On A Graph </h2>
<p>So let us see a few examples to understand what is going on. </p>
<p>When <b>A</b> and <b>B</b> are subsets of the Real Numbers we can graph the relationship. </p>
<p>Let us have <b>A</b> on the x axis and <b>B</b> on y, and look at our first example:</p>
<p class="center"><img src="images/vertical-line-test.svg" alt="function not single valued"></p>
<p>This is <b>not a function</b> because we have an <b>A</b> with many <b>B</b>. It is like saying f(x) = 2 <i><b>or</b></i> 4 </p>
<p>It fails the "Vertical Line Test" and so is not a function. But is still a valid relationship, so don't get angry with it.</p>
<p>Now, a general function can be like this:</p>
<p class="center"><img src="images/function-general-graph.svg" alt="General Function"><br>
<span class="larger">A General Function</span></p>
<p>It CAN (possibly) have a <b>B</b> with many <b>A</b>. For example sine, cosine, etc are like that. Perfectly valid functions.</p>
<p>But an "<b>Injective Function</b>" is stricter, and looks like this:</p>
<p class="center"><img src="images/function-injective-graph.svg" alt="Injective Function"><br>
<span class="larger">"Injective" (one-to-one)</span></p>
<p>In fact we can do a "Horizontal Line Test": </p>
<div class="def">
<p>To be <b>Injective</b>, a Horizontal Line should never intersect the curve at 2 or more points.</p>
</div>
<p><i>(Note: <a href="functions-increasing.html">Strictly Increasing (and Strictly Decreasing) functions</a> are Injective, you might like to read about them for more details)</i></p>
<p>So:</p>
<ul>
<li>If it passes the <b>vertical line test</b> it is a function</li>
<li>If it also passes the <b>horizontal line test</b> it is an injective function</li>
</ul>
<h2>Formal Definitions</h2>
<p>OK, stand by for more details about all this: </p>
<h3>Injective </h3>
<p>A function <b><i>f</i></b> is <b>injective</b> if and only if whenever <b><i>f(x) = f(y)</i></b>, <b><i> x = y</i></b>. </p>
<div class="example">
<p><b>Example:</b> <b><i>f</i>(<i>x</i>) = <i>x+5</i></b> from the set of real numbers <img src="../images/symbols/set-r.svg" alt="real numbers" height="16"> to <img src="../images/symbols/set-r.svg" alt="real numbers" height="16"> is an injective function. </p>
<p>Is it true that whenever <b><i>f(x) = f(y)</i></b>, <b><i> x = y</i></b> ?</p>
<p>Imagine x=3, then:</p>
<ul>
<li> f(x) = 8</li>
</ul>
<p>Now I say that f(y) = 8, what is the value of y? It can only be 3, so x=y </p>
</div>
<br>
<div class="example">
<p><b>Example:</b> <b><i>f</i>(<i>x</i>) = <i>x<sup>2</sup></i></b> from the set of real numbers <img src="../images/symbols/set-r.svg" alt="real numbers" height="16"> to <img src="../images/symbols/set-r.svg" alt="real numbers" height="16"> is <b class="large">not</b> an injective function because of this kind of thing:</p>
<ul>
<li><b><i>f</i>(<i>2</i>) =
4 </b>and </li>
<li><b><i>f</i>(<i>-2</i>) = 4</b></li>
</ul>
<p>This is against the definition <b><i>f(x) = f(y)</i></b>, <b><i> x = y</i></b>, because <i><b>f(2) = f(-2) but 2 ≠ -2</b></i></p>
<p>In other words there are <b>two</b> values of <b>A</b> that point to one <b>B</b>.</p>
<p>&nbsp;</p>
<p>BUT if we made it from the set of natural
numbers <img src="../images/symbols/set-n.svg" alt="natural numbers" height="16"> to <img src="../images/symbols/set-n.svg" alt="natural numbers" height="16"> then it <span class="large">is</span> injective, because:</p>
<ul>
<li><b><i>f</i>(<i>2</i>) =
4 </b></li>
<li>there is no f(-2), because -2 is not a natural
number</li>
</ul>
<p>So the domain and codomain of each set is important!</p>
</div>
<p>&nbsp;</p>
<h3>Surjective (Also Called "Onto")</h3>
<p> A function <b><i>f</i></b> (from set <i><b>A</b></i> to <i><b>B</b></i>) is <b>surjective</b> if and only if for every <b>
<em>y</em></b> in <i><b>B</b></i>, there is at least one <b><em>x</em></b> in <i><b>A</b></i> such that <span class="style1"><em>f</em>(<em>x</em>) = <em>y</em></span>,<b> </b>in other words &nbsp;<b><i>f</i></b> is surjective
if and only if
&nbsp;<b><i>f(A) = B</i></b>. </p>
<p>In simple terms: every B has some A. </p>
<div class="example">
<p><b>Example:</b> The function <b><i>f</i>(<i>x</i>) = <i>2x</i></b> from the set of natural
numbers <img src="../images/symbols/set-n.svg" alt="natural numbers" height="16"> to the set of non-negative <b>even</b> numbers is a <b>surjective</b> function. </p>
<p>BUT <b><i>f</i>(<i>x</i>) = <i>2x</i></b> from the set of natural
numbers <img src="../images/symbols/set-n.svg" alt="natural numbers" height="16"> to <img src="../images/symbols/set-n.svg" alt="natural numbers" height="16"> is <b>not surjective</b>, because, for example, no member in <img src="../images/symbols/set-n.svg" alt="natural numbers" height="16"> can be mapped to <b><i>3</i></b> by this function. </p>
</div>
<p>&nbsp;</p>
<h3>Bijective</h3>
<p>
A function <b><em>f</em></b> (from set <i><b>A</b></i> to <i><b>B</b></i>) is <b>bijective</b> if, for every <b><em>y</em></b> in <i><b>B</b></i>, there is exactly one <b><em>x</em></b> in <i><b>A</b></i> such that <span class="style1"><em>f</em>(<em>x</em>) = <em>y</em></span></p>
<p>Alternatively, <b><em>f</em></b> is bijective if it is a <strong>one-to-one correspondence</strong> between those sets, in other words both <strong>injective and surjective.</strong></p>
<div class="example">
<p>
<b>Example:</b> The function <b><i>f</i>(<i>x</i>) = <i>x<sup>2</sup></i></b> from the set of positive real
numbers to positive real
numbers is both injective and surjective.
Thus it is also <b>bijective</b>. </p>
<p>But the same function from the set of all real numbers <img src="../images/symbols/set-r.svg" alt="real numbers" height="16"> is <span class="large">not</span> bijective because we could have, for example, both</p>
<ul>
<li><b><i>f</i>(<i>2</i>)=4 and </b></li>
<li><b><i>f</i>(<i>-2</i>)=4</b></li>
</ul>
</div><p>&nbsp;</p>
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