Types Of Relations In Math | TutorsOnNet
In this blog post we'll be studying relations between sets. A relation called R on set A is reflexive if for every ordered pair, (x, x) is an element. Date: 11/10/98 at From: Mike Subject: Discrete math Suppose R is a Suppose R is a reflexive and transitive relation on A. Define a new relation. Reflexive. A relation is reflexive if, we observe that for all values a: In a reflexive relation, we have arrows for all values in the.
Relations and functions
The main reason for not allowing multiple outputs with the same input is that it lets us apply the same function to different forms of the same thing without changing their equivalence. This is the same as the definition of function, but with the roles of X and Y interchanged; so it means the inverse relation f-1 must also be a function. In general—regardless of whether or not the original relation was a function—the inverse relation will sometimes be a function, and sometimes not.
When f and f-1 are both functions, they are called one-to-one, injective, or invertible functions. In other words, a surjective function f maps onto every possible output at least once. A function can be neither one-to-one nor onto, both one-to-one and onto in which case it is also called bijective or a one-to-one correspondenceor just one and not the other. Relations[ edit ] In the above section dealing with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.
Writing in set notation, if a is some fixed value: In other words, the number of outputs that a function f may have at any fixed input a is either zero in which case it is undefined at that input or one in which case the output is unique. However, when we consider the relation, we relax this constriction, and so a relation may map one value to more than one other value.
In general, a relation is any subset of the Cartesian product of its domain and co-domain. It could be either one. So you don't have a clear association. If I give you 1 here, you're like, I don't know, do I hand you a 2 or 4?
That's not what a function does. A function says, oh, if you give me a 1, I know I'm giving you a 2. If you give me 2, I know I'm giving you 2. Now with that out of the way, let's actually try to tackle the problem right over here.
So let's think about its domain, and let's think about its range. So the domain here, the possible, you can view them as x values or inputs, into this thing that could be a function, that's definitely a relation, you could have a negative 3. You could have a negative 2. You could have a 0. You could have a, well, we already listed a negative 2, so that's right over there. Or you could have a positive 3. Those are the possible values that this relation is defined for, that you could input into this relation and figure out what it outputs.
Now the range here, these are the possible outputs or the numbers that are associated with the numbers in the domain.
The range includes 2, 4, 5, 2, 4, 5, 6, 6, and 8. I could have drawn this with a big cloud like this, and I could have done this with a cloud like this, but here we're showing the exact numbers in the domain and the range. And now let's draw the actual associations. So negative 3 is associated with 2, or it's mapped to 2. So negative 3 maps to 2 based on this ordered pair right over there. Then we have negative 2 is associated with 4. So negative 2 is associated with 4 based on this ordered pair right over there.
Actually that first ordered pair, let me-- that first ordered pair, I don't want to get you confused. It should just be this ordered pair right over here. Negative 3 is associated with 2. Then we have negative we'll do that in a different color-- we have negative 2 is associated with 4. Negative 2 is associated with 4. We have 0 is associated with 5.
Or sometimes people say, it's mapped to 5. We have negative 2 is mapped to 6. Now this is interesting. Negative 2 is already mapped to something. Now this ordered pair is saying it's also mapped to 6.
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And then finally-- I'll do this in a color that I haven't used yet, although I've used almost all of them-- we have 3 is mapped to 8. So the question here, is this a function? And for it to be a function for any member of the domain, you have to know what it's going to map to.
It can only map to one member of the range. So negative 3, if you put negative 3 as the input into the function, you know it's going to output 2. If you put negative 2 into the input of the function, all of a sudden you get confused.
Do I output 4, or do I output 6? So you don't know if you output 4 or you output 6.
And because there's this confusion, this is not a function. You have a member of the domain that maps to multiple members of the range.
So this right over here is not a function, not a function.