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Hello.
I have the question:
Let A = {a, b, c, x, y, z} Give an example of a relation R on A which is:
(a) reflexive and not transitive,
(b) transitive and not reflexive,
(c) transitive and not symmetric,
(d) antisymmetric, transitive, but not reflexive,
(e) an equivalence relation,
(f) an order relation.
(g) a function.
In my professors notes he never does any examples that have so many elements in the set, and therefore uses them all in every example problem.
Do I have to use every element in the set A for the relation R?
Or can I just say that the answer to (a) would be something like {(a,a), (a,b), (b,b)}
Also, I don't really understand how to do the function question, any explanation would be quite helpful.
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Or can I just say that the answer to (a) would be something like {(a,a), (a,b), (b,b)}
That is the form you want your answer in. But that answer is not correct. It is both not reflexive (is c related to c?) and transitive.
Also, I don't really understand how to do the function question, any explanation would be quite helpful.
What is the definition of a function? (Hint: Use the word "relation" in your definition).
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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So from what Ricky said the answer to (a), would be {(a,a), (b,b), (c,c), (x,x),(y,y),(z,z)}
For (b), for the relation to be transitive is it something like {(a,b), (b,c), (a,c),(b,a),(b,a),(c,a)(c,b).....(z,a),(z,b),(z,c)}
Is their a shorter answer, do I need to use every element from the set A in the relation example? I'm horribly confused.
I'm not really getting this very well, I don't fully understand what constitutes the relation. I only have one example of each
type of the relations and our class isn't really on this. The book for the course doesn't cover this at all so I just have some small notes of the prof's. (I'm actually missing a prerequisite for the course)
I also have basically the same with a different set.
Let A = Z Give an example of a relation R on A which is:
(a) reflexive and not transitive,
(b) transitive and not reflexive,
(c) transitive and not symmetric,
(d) antisymmetric, transitive, but not reflexive,
(e) an equivalence relation,
(f) an order relation.
(g) a function.
For answers for this, I don't really know what to do?
I feel like I'm missing a lot on this, does anyone know of a good web resource on set relations?
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So from what Ricky said the answer to (a), would be {(a,a), (b,b), (c,c), (x,x),(y,y),(z,z)}
That is not correct. Transitive means that IF g is related to h, and h related to i, then g is related to i. However, if nothing is ever related to each other, then you have that this statement is vacuously true.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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So in order for them not to be transitive, they have to be at least related? I almost feel like you misread the question, or I'm really not getting this at all.
For it to be reflexive and transitive(which is not the question though): {(a,a), (b,b), (c,c), (x,x),(y,y),(z,z), (a,z),(z,a)}
For it to be transitive and non-reflexive: {(a,a), (b,b), (c,c), (x,x),(y,y),(a,z),(z,a)}
That makes sense to me I believe.
Would that be right now?
Thanks Ricky.
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So in order for them not to be transitive, they have to be at least related?
Think of it this way: In order for it not to be transitive, there must exist a counter example to the transitive property. So what does it take for something to be a counter example? You have to satisfy the hypothesis, and the conclusion can not hold. The hypothesis for transitive is that g is related to h and h is related to i. The conclusion is that g is related i.
If you have {(a,a), (b,b), (c,c)}, that set is transitive. The only elements that are related is (a,a), (b,b), and (c,c). So the only thing that satisfies the hypothesis is a is related to a, and a is related to a (or the same thing with all b's and c's). And of course, a is related to a, so the conclusion holds as well. So for everything that satisfies the hypothesis of transitive, the conclusion holds. Therefore, there are no counter-examples and so the set is transitive.
For it to be reflexive and transitive(which is not the question though): {(a,a), (b,b), (c,c), (x,x),(y,y),(z,z), (a,z),(z,a)}
That relation is indeed reflexive and transitive.
For it to be transitive and non-reflexive: {(a,a), (b,b), (c,c), (x,x),(y,y),(a,z),(z,a)}
That set is not transitive. z is related to a, and a is related to z, but z is not related to z. So we have satisfied they hypothesis, but the conclusion of transitive does not hold. Therefore, we have found a counter example to the property of transitive.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Thank you Ricky, that makes so much more sense now.
If the set the relation was on was all integers how would I go about writing that as a relation?
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If the set the relation was on was all integers how would I go about writing that as a relation?
As with any thing that is infinite, we may define functions or recursive algorithms. For example:
n is related to m if and only if n = m/2.
So we see that 4 is related to 8, but 8 is not related to 4, and so on.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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