Reflexive, symmetric, transitive relations

Nov 2014
27
1
math
Let A = {1, 2, 3}
Define a relation R on A that is:

not reflexive, not symmetric, transitive
my ans: {(1,2), (2,3), (1,3)}
Can it be only three elements?

not reflexive, symmetric, transitive
my ans: {(1,2), (2,1), (2,3), (3,2), (1,3), (3,1)}
I read that each of the three property should be independent to each other, but in this case if the relation is transitive, (1,2) and (2,1) will imply (1,1) is in R. But it is given that R is not reflexive. What should be the right answer?

reflexive, symmetric, not transitive
my ans: {(1,1), (2,2), (3,3)}
Correct? Same as the relation that is reflexive, not symmetric and not transitive?

not reflexive, not symmetric, not transitive
my ans: {(1,2), (1,3)}
Correct?

reflexive, symmetric, transitive
my ans: {(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)}
Correct?
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
Let A = {1, 2, 3}
Define a relation R on A that is:

not reflexive, not symmetric, transitive
my ans: {(1,2), (2,3), (1,3)}
Can it be only three elements?
Yes, of course. Why would you have to ask?

not reflexive, symmetric, transitive
my ans: {(1,2), (2,1), (2,3), (3,2), (1,3), (3,1)}
I read that each of the three property should be independent to each other, but in this case if the relation is transitive, (1,2) and (2,1) will imply (1,1) is in R. But it is given that R is not reflexive. What should be the right answer?
Yes, this is not a good example- it is not transitive because, as you say, since it contains (1, 2) and (2, 1) to be transitive it must contain (1, 1). However, just containing (1, 1) is NOT enough to make it "reflexive". To be reflexive, the set must contain all three of (1, 1), (2, 2), and (3, 3). If you add (1, 1) and (2, 2) but not (3, 3) to what you have, it will be symmetric and transitive but not reflexive.

reflexive, symmetric, not transitive
my ans: {(1,1), (2,2), (3,3)}
Correct? Same as the relation that is reflexive, not symmetric and not transitive?
No, this is not correct because it is transitive. Transitive means
"if (a, b) and (b, c) are in the relation then so is (a, c)". Here the only "(a, b)" and "(b, c)" would be "(1, 1) and (1, 1) so (1, 1) must be in the set", "(2, 2) and (2, 2) so (2, 2) must be in the set", and "(3, 3) and (3, 3) so (3, 3) must be in the set", all of which are true.

not reflexive, not symmetric, not transitive
my ans: {(1,2), (1,3)}
Correct?
Yes, that is correct.

reflexive, symmetric, transitive
my ans: {(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)}
Correct?[/QUOTE]
Yes, that is correct. An even simpler example would be {(1, 1), (2, 2), (3, 3)}.
Are you familiar with "equivalence relations"? Your example is essentially saying that all three of 1, 2, and 3 are equivalent. My example says that 1 is equivalent to 1 only, 2 is equivalent to 2 only, and 3 is equivalent to 3 only.
 
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Nov 2014
27
1
math
Thanks for clearing up my misconceptions!!!!

not reflexive, not symmetric, transitive
my ans: {(1,2), (2,3), (1,3)}
Can it be only three elements?
Yes, of course. Why would you have to ask?
I thought all the elements in A has to be included in both "coordinates". Now I know that's only true for reflexivity.

not reflexive, symmetric, transitive
{(1,2), (2,1), (2,3), (3,2), (1,3), (3,1), (1,1), (2,2)}
but since (3,2) and (2,3) are in R, (3,3) should be there...
{(1,1), (1,2), (2,1)}
I believe this is correct.

reflexive, symmetric, not transitive
my ans: {(1,1), (2,2), (3,3), (1,2), (2,3), (2,1), (3,2)}
not transitive since (1,3) is not in R.
 

CRGreathouse

Forum Staff
Nov 2006
16,046
936
UTC -5
not reflexive, not symmetric, transitive
my ans: {(1,2), (2,3), (1,3)}
Can it be only three elements?
Looks good.

not reflexive, symmetric, transitive
my ans: {(1,2), (2,1), (2,3), (3,2), (1,3), (3,1)}
I read that each of the three property should be independent to each other, but in this case if the relation is transitive, (1,2) and (2,1) will imply (1,1) is in R. But it is given that R is not reflexive. What should be the right answer?
This isn't transitive, as you explained, so you need to try something else.

reflexive, symmetric, not transitive
my ans: {(1,1), (2,2), (3,3)}
Correct? Same as the relation that is reflexive, not symmetric and not transitive?
This relation is transitive; you need some (x, y) and (y, z) in the relation such that (x, z) is not in the relation, but no such x, y, z exist at present. (Clearly, if (x, y) was in the current relation, x = y and so you're looking for some (x, z) with x != z, but that doesn't happen.)

not reflexive, not symmetric, not transitive
my ans: {(1,2), (1,3)}
Correct?
No, this is transitive. You need some (x, y) and (y, z) in the relation such that (x, z) is not in the relation.

reflexive, symmetric, transitive
my ans: {(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)}
Correct?
Yes.
 
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CRGreathouse

Forum Staff
Nov 2006
16,046
936
UTC -5
not reflexive, symmetric, transitive
{(1,2), (2,1), (2,3), (3,2), (1,3), (3,1), (1,1), (2,2)}
This is not transitive, since it has (3,1) and (1,3) but not (3,3). You'll need to remove more elements.

reflexive, symmetric, not transitive
my ans: {(1,1), (2,2), (3,3), (1,2), (2,3), (2,1), (3,2)}
not transitive since (1,3) is not in R.
Yes.
 
Jun 2014
945
191
Earth
Let A = {1, 2, 3}
Define a relation R on A that is:

not reflexive, not symmetric, transitive
my ans: {(1,2), (2,3), (1,3)}
Can it be only three elements?
Country Boy said:
Yes, of course. Why would you have to ask?
Country Boy, there is no "of course" about it. If it was "of course," then
the student wouldn't be asking it in the first place. Think about what you are typing.
It's not logical.
 
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