Surjective (onto) and injective (one-to-one) functions | Linear Algebra | Khan Academy

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In this video, I want to present to you a useful term in our discussions about functions and the ability to find the inverse of functions. In general, you will encounter this term a lot while working as an athlete. So let’s say I have the conjugate f (f), which is converted from group x (x) to group y (wy). We’ve drawn this illustration several times, but it doesn’t hurt to draw it again. This is the group X or domain. This group is Y or the corresponding field. Remember, the corresponding field is the group that you are converting to. It is not necessary to convert to each element of the group These are all the elements, the group that you can convert to in the corresponding field ## Let’s see If I have something here, f will convert it to another object corresponding to it in the corresponding field Y. So the first idea, or term, that I want to present to you is the idea that conjugation is universal. And sometimes this is called onto ### The conjugation is universal if it is any element in the corresponding field – I will write it like this – for each, say y, which belongs to the opposite range Y, there is – this is a shortened sign of the phrase “there” – there is at least x that belongs to the field X, where We can write that way this way Frankly, let’s write the same word. Where the conjugation f of the variable x is equal to y. It is important to say, you can choose any y value here. And all y here is converted to at least one of the x values \u200b\u200bhere. So, for example, let’s draw a simpler example instead of all of these writings. Let’s say I have a group Y that looks exactly like this. Say, group Y has only 4 elements The group contains both a, b, c and d (e, b, c, and d) This is the Y group. And let’s say group X looks like this. Let’s say elements 1, 2, 3, and 4 belong to group x. Now, for the conjugation f to be inclusive, each of these elements must be transferable here. # What do we mean by that? If we could connect each of these elements, let’s set another example. Let’s say this turns into that. And let’s say that I’m going to draw a fifth element here exactly, let’s say that both of these elements are related to d So the coupling f of element 4 is d, and so does 5 This is an example of universal pairing So these elements represent the values \u200b\u200btransformed by the conjugation f here. The conjugation f is an inclusive conjugation What do we mean by that? By this we mean that each element here is converted to another in the corresponding field. Now, I’ll give you an example of a coupling on a non-exhaustive coupling. I will add more elements to group Y. We will add element e which represents one of the y-values. Suddenly, we see that the conjugation f is not exhaustive. Why? The reason is that an element of group Y has not been converted to. The meaning of the conjugation f inclusive is that the conjugation represents the transformation of all values, that all values \u200b\u200bbe converted to one value on the bottom here. So, you can, all of the elements somehow represent one to one conversion. ## And I will know it better in the coming days. So, it could be like this, and so on. ## We can also connect two of this group to one of the other But the basic requirement is that everything here is transferred from the corresponding shop to the other Another way to visualize it; What if we take the picture? So, universal pairing – I’ll write it here Let’s write it like this, if we say that f is inclusive, that means that there is a picture of the conjugation f. Remember that the image was, all the values \u200b\u200bthat f converts to .. This means that f is equal to y. Now, I learned before, that the image does not have to equal the corresponding field However, if the conjugation is comprehensive, then this means that the picture equals the opposite field. Everything that is in the corresponding field will be transferred to it Another name for the image is the pairing range. You can also say that the range f is equal to Y. Remember the differences, as I drew a distinction where we first spoke about functions, the distinction between the opposite field and the range, the opposite range is a group that you can convert to. You don’t have to convert every element here. A range is a subset of the corresponding field to which it is converted. If you have to estimate the coupling output for each of these points, you are already converting to the range, which we also call the image. The word image used more commonly within linear algebra. But if the image or range is equal to the opposite field, if all elements in the corresponding field are converted to, then you are dealing with a comprehensive conjugation Now, the next term I’m going to introduce you to is the contrast pairing. This pairing is a one-to-one coupling Let’s draw the corresponding field and field again So let’s say this is the field, and this is the opposite field. This is X and this is Y If we say that f is a differential coupling, or one-to-one coupling, this means that for each of the values \u200b\u200bto be converted to, I will write it like this, for each value connected to it – let’s say, I will say it in a number of different ways – there is at most one x to convert to. Or, in other words, for each y belongs to the Y group – to write it this way – for any y that belongs to Y, there is at most one of the X values \u200b\u200bthat are converted to it, since the conjugation f of the variable x is equal to y. There may be no x values \u200b\u200bthat you can convert to. For example, you could take an element of Y here and which is never converted to it. Other than that, the rest of the items are converted to it except this. Thus, this could be the case where you do not realize the principle of universal pairing. This conjugation is not universal since this element, which belongs to the corresponding field, does not belong to the image or the extent. It cannot be converted to. This would not be a variation when the X values \u200b\u200bwere converted to single and distinct values \u200b\u200bin Group Y. Now, how can a pairing not be a contrast or a one-to-one? I think you understand the principle that conjugation should be one to one. Well, if two values \u200b\u200bfrom group X are converted to one y, or even 3 turns to the same y value, this means that we’re not dealing with a differential coupling. So, this is what we mean by contrast. Let’s take another example here. Rather, let’s return to this example here. When we added the variable e here, we said the conjugation is not universal. Because we did not convert to each of these elements. However, is conjugation contrasting? No, because f of the 4 and 5 values \u200b\u200bare converted to d. Thus, this is what nullifies the hypothesis of whether or not conjugation is heterogeneous. This also denies the inclusion of coupling. Now, if I want to write a national and comprehensive conjugation, I have to erase this transformation and change the 5 conjugation to e Now, all values \u200b\u200bare converting from one to one, and I don’t have a conversion from two values \u200b\u200bfrom X to one from Y. Plus all Y values \u200b\u200bare converted to Consequently the conjugation has become one-to-one and universal.

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