3 Bijectivity and Inverses Now, let's consider the situation with injectivity and surjectivity pictorially When we say a function is injective, what we mean is that if
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[PDF] A function is bijective if and only if has an inverse
30 nov 2015 · We say that f is injective if whenever f(a1) = f(a2) for some a1,a2 ∈ A, then a1 = a2 We say that f is bijective if it is both injective and surjective Let f : A → B be bijective Then f has an inverse
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[PDF] Math 127: Functions
3 Bijectivity and Inverses Now, let's consider the situation with injectivity and surjectivity pictorially When we say a function is injective, what we mean is that if
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Math 127: Functions
Mary Radclie
1 Basics
We begin this discussion of functions with the basic denitions needed to talk about functions. Denition 1.LetXandYbe sets. AfunctionffromXtoYis an object that, for each elementx2X, assigns an elementy2Y. We use the notationf:X!Yto denote a function as described. We write f(x) =yorf:x7!yto denote that the element inYassigned toxisy. We callXthedomainoff, and we callYthecodomainoff. Iff(x) =y, we say thatxmaps toyunderf.In general, we will often talk about functions from this perspective of \mapping;" we see the role of
a function as taking things fromX, and sending them over toY, where the functionfis the map thatexplains where each element inXis supposed to go. This is visualized in Figure 1.; Let's think about this notion with a function we probably feel a little more comfortable with. Consider asking us to do is map elements inZover to new elements inZ, where the directions on the map, described by the function, are to multiply something by 2. So if you ask the function \what do I do with the number Now, in order for this map to make sense, there are a few basic properties that we need to ensure are While these properties are trivially true in the denition of a function, it's useful to think about the failure cases to think about what kinds of assignments we might make that are in fact NOT functions. In general, if you are to dene a function yourself, it's worth thinking about these things to ensure that the function you are creating is a well-dened function.Example 1.Deneg:R!Rbyg(x) =1x1. This is NOT a well-dened function, becauseg(1) Just to be sure, let's prove that 0:9999= 1, which in turn clearly implies that 0:4999= 0:5. First, leta= Since 9a= 9, we therefore have thata= 1. Thus, 1 =a= 0:99999:QED.When you are dening a function, be on the lookout for these kinds of problems. Some things to think As with several of our previous topics, we have some special functions that will show up from time to produce the same value. However, we cannot say that these are really the same function, because there For these reasons, we distinguish between identical equality of functions and equality on elements. This We have, already, some language and notation to discuss things like the domain or codomain of a function. Yare assigned to elements in this subset? This is precisely what the image ofUis getting at.Example 4.Denef:Z!Nbyf(x) =jxj+1. To illustrate the concept of image, let's consider containing only one elementxis the set containing only one element,f(x).In the case of the function shown in Example 4, we have that the range of the functionfis precisely the same as the codomain off. This is because for this function, every element in the codomain is the dened byg(z) = 2z, we clearly have that the codomain is all ofZ, but the range is just the even integers. elements inXmight map to that subset. In that case, we can dene a similar type of object to the image, ensure we understand the concept, let's consider an example.Example 5.Letf:Z!Zwithf(z) =j2zjfor allz2Z. Let's consider the preimage of a few Hence, we havef1(V) =Z.Unlike with the image, as we see above the number of elements in the preimage of a setVis not We shall not prove all these properties here; those that we do not prove will be left as an exercise. In function can disregard the setYcompletely, as shown in Figure 3. The notation for function composition is set up so that the rst function you perform is closest to thex;gfindicates thatfhas to be performed3?" the function will answer \Send it over to 6." In this way, the function provides us with a mechanism
to transform the number 3 according to some rule, which is the denition of the function. Figure 2: A representation of the functionf:Z!Zdened byf(n) = 2n. Existence: For everyx2X,f(x)2Y.
Uniqueness: For everyx2X, there is only oney2Ysuch thatf(x) =y. Indeed
a, 0:5 = 0:4999999:::, and hence we have more than oney2 f0;1;:::;9gwithf(12 ) =y. This assignment of values fails the uniqueness property.a 0:9999:::, which is some real number. Then 10a= 9:9999:::, and hence 9a= 10aa= 9:99990:9999= 9.
8x2X; f(x) =x.
Theempty functionis any functionf:; !X. Note that there is no need, in the empty function, to dene any values for elements in the domain, as there are none! Finally, we have to address the question of what it means for two functions to be the same. Denition 4.LetX;Y;A;Bbe sets, and letf:X!Yandg:A!Bbe functions. We say thatfis identically equal tog, denoted byfg, if the following conditions are met: X=A Y=B 8 x2X,f(x) =g(x).
There is a reason for distinguishing equality of functions withrather than =. There are often times, for example, when it is important to nd a specic choice ofx2Xfor whichf(x) =g(x). However, this is quite dierent fromf(x) =g(x) for every singlex2X. Moreover, we can have functions that are the same on all their common domain elements (that is, f(x) =g(x)8x2X\A), but are not identically equal because perhaps they have dierent domains. An example might be as follows: denef:N!Nbyf(x) = 3x, and forAthe set of even integers, dene g:A!Zbyg(a) = 3a. Then on anyxthat is both a natural number and an even integer,fandg 1.1 Functions and Subsets
1(V) =fx2Xjf(x)2Vg:
This is to say, the preimage ofVis the set of all elements inXwhose image is a member ofV. To 1(V) =f1;1g.
IfV=f1g, then the preimage ofVis all those elements inZthat map to 1; that is, it is all choices ofzfor whichf(z) =j2zj= 1. There are no such elements! Hence,f1(V) =;. IfV=f0g, then the preimage ofVis all those elements ofZthat map to 0, which is clearly justz= 0. Hencef1(V) =f0g. IfV=f0;1;2;3;4g, then by repeating above ideas, we have that the elements inZfor which f(z)2Vare exactlyf1(V) =f2;1;0;1;2g. If we takeV=fx2Zjx0 andxis eveng, then every element ofZhas its image inV. If U1;U2X, thenf(U1[U2) =f(U1)[f(U2).
2. If U1;U2X, thenf(U1\U2)f(U1)\f(U2).
3. If V1;V2Y, thenf1(V1[V2) =f1(V1)[f1(V2).
4. If V1;V2Y, thenf1(V1\V2) =f1(V1)\f1(V2).
3.W epro vethe set e qualityb ydouble con tainment.
First, suppose thatx2f1(V1[V2). Then by denition,f(x)2V1[V2, so eitherf(x)2V1orf(x)2 V 2. Wolog, suppose thatf(x)2V1. Then we havex2f1(V1), and hencex2f1(V1)[f1(V2).
For the other direction, suppose thatx2f1(V1)[f1(V2). Then we havex2f1(V1) orx2 f 1(V2); wolog suppose thatx2f1(V1). Then by denition, we havef(x)2V1, and thus
f(x)2V1[V2. But thenx2f1(V1[V2). Therefore, we have thatf1(V1[V2) =f1(V1)[f1(V2) by double containment. In addition, we have similar properties for set complements. Proposition 2.LetX;Ybe sets, and letf:X!Ybe a function. 1. If UX, thenf(XnU)f(X)nf(U).
2. If VY, thenf1(YnV) =Xnf1(V)
The proofs of these properties are left as an exercise. 1.2 Composition
Denition 8.LetX;Y;Zbe sets, and letf:X!Yandg:Y!Zbe functions. We dene the compositiongf:X!Zto be the function dened bygf(x) =g(f(x)). We can imagine a composition as follows. The functionfgives us a rule for how to map the setXinto the setY, and the functionggives us a rule for how to map the setYinto the setZ. By chaining these rules together, rst by doing thefrule, then thegrule, we can get fromXall the way toZ. This new