Thank you! A lot of videos I have come across failed to mention the point that it must contain ALL (x,x). It might sound stupid but your video is one of the best explanations!
This is the simplest explanation of relation i've seen so far in my surf through youtube for added knowledge on relations. But I have a question What if R5 = A x A, will it still be a reflexive relation? If yes, what are the reasons. If no, then why? Thank you. Expecting your reply.
Hi Aaron. Yes. If the relation is actually the whole cross product then it will contain all (x, y) pairs, and in particular, all (x, x) pairs. Thus, it will be reflexive. Regards. Jonathan
Hi Abdlmalek. Yes, it is still reflexive. The test of reflexivity is only concerned with all 'x in A' that there is a '(x, x) in R.' So if that is true then irrespective of the others it is still reflexive. For example. If A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 4)}, this is reflexive even thou (2, 4) is present. Regards. Jonathan.
Hi Abhi, Yes, that makes sense. Given a set A with cardinality |A| = n, all relations are subsets of the cross-product AxA. So, the total number of relations that you can create is the number of possible subsets of AxA. All subsets of AxA is given by the power set of AxA, P(A). Its cardinality |P(A)| = 2^|AxA|. Now if A has cardinality n, then any reflexive relation must contain the n ordered pairs of the form (x, x) from AxA. Now consider AxA without those n ordered pairs, this set contains n^2 - n ordered pairs. Generate all subsets of that set and place in each the n reflexive couples, to make them all reflexive. So given a set A of cardinality |A| = n, the total number of reflexive relations on A is given by: 2^(n^2 - n). With respect to the example in the video. |A| = 3, therefore |AxA| = 9. The number of reflexive relations is 2^(9-3) = 2^6 = 64. As you suggested. I hope this helps.. Regards. Jonathan.
Hi, I have a question, if the relation contains only 2 of (x,x) e.g. (1,1),(2,2) is it still reflexive? Or does it need all 3 elements - (1,1),(2,2) and (3,3)?? Thanks
Hi Iris, assuming the set that the relation was constructed on is {1, 2, 3} the all three ordered pairs (1, 1), (2, 2), and (3, 3) must be present for the relation to be reflexive. If it only contains (1, 1) and (2, 2) then it is not reflexive. Regards. Jonathan
Hi Andries, I suppose the short answer to your question is no, just because a relation is not reflexive doesn't necessarily make it irreflexive. Let us recall the definitions: A relation is reflexive only if for all 'x' in our base set we have (x, x) in the relation. Key words here being 'for all.' A relation is irreflexive only if for all 'x' in our base set we have (x, x) is not in the relation. Key words here being 'for all.' So the relation R1 above is not reflexive as (4, 4) and (7, 7) are missing, but R1 is also not irreflexive as (2, 2) is present. Relation R3 is not reflexive as (2, 2), and (4, 4), and (7, 7) are missing, now because they are all missing it is now irreflexive. R4 is an unusual case, it is the empty set, and so is vacuously reflexive and also irreflexive, it would be the only example of a set that is both. I hope this helps. Regards Jonathan
This was an absolutely incredible explanation! This truly deserves far more views!
Hi Liron, Thanks for your kind words. Please share. Regards. Jonathan.
Will do!
Thank you! A lot of videos I have come across failed to mention the point that it must contain ALL (x,x). It might sound stupid but your video is one of the best explanations!
Yea, I searched billion pages they all suck. They just give the definition and expecting us to understand foolish bullshit. This is great explanation.
I agree with others here, this is the best explanation. Thank you so much!
i owe you big time. thank you
Very well explained. Good work sir
thank you for the explanation. have a nice day
Thank you so much for this explanation!
Thank you for this easy explanation
Dont undestand shit from my class but this def made it so even my small brain could understand. Thank you sir bless ur heart
عاشت ايدك ضلعي شرحك فدشي 🌚❤️
This is the simplest explanation of relation i've seen so far in my surf through youtube for added knowledge on relations. But I have a question
What if R5 = A x A, will it still be a reflexive relation? If yes, what are the reasons. If no, then why?
Thank you. Expecting your reply.
Hi Aaron.
Yes. If the relation is actually the whole cross product then it will contain all (x, y) pairs, and in particular, all (x, x) pairs. Thus, it will be reflexive.
Regards.
Jonathan
one question: what if the relation contains (x,x) for every x in A but also has other elements, is it still reflective?
Hi Abdlmalek. Yes, it is still reflexive. The test of reflexivity is only concerned with all 'x in A' that there is a '(x, x) in R.' So if that is true then irrespective of the others it is still reflexive.
For example. If A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 4)}, this is reflexive even thou (2, 4) is present.
Regards. Jonathan.
Sir, I read 2^nsquare - n is used to find the number of reflexive relations. As per ur example number of reflexive relatives to be 64 right ?
Hi Abhi,
Yes, that makes sense. Given a set A with cardinality |A| = n, all relations are subsets of the cross-product AxA. So, the total number of relations that you can create is the number of possible subsets of AxA.
All subsets of AxA is given by the power set of AxA, P(A). Its cardinality |P(A)| = 2^|AxA|.
Now if A has cardinality n, then any reflexive relation must contain the n ordered pairs of the form (x, x) from AxA. Now consider AxA without those n ordered pairs, this set contains n^2 - n ordered pairs. Generate all subsets of that set and place in each the n reflexive couples, to make them all reflexive.
So given a set A of cardinality |A| = n, the total number of reflexive relations on A is given by:
2^(n^2 - n).
With respect to the example in the video. |A| = 3, therefore |AxA| = 9. The number of reflexive relations is 2^(9-3) = 2^6 = 64.
As you suggested.
I hope this helps.. Regards. Jonathan.
Hi,
I have a question, if the relation contains only 2 of (x,x) e.g. (1,1),(2,2) is it still reflexive?
Or does it need all 3 elements - (1,1),(2,2) and (3,3)??
Thanks
Hi Iris, assuming the set that the relation was constructed on is {1, 2, 3} the all three ordered pairs (1, 1), (2, 2), and (3, 3) must be present for the relation to be reflexive. If it only contains (1, 1) and (2, 2) then it is not reflexive. Regards. Jonathan
thank you :)
Another question since in you example R1,3 and 4 are not reflexive does that make them irreflexive by definition.
Hi Andries, I suppose the short answer to your question is no, just because a relation is not reflexive doesn't necessarily make it irreflexive. Let us recall the definitions:
A relation is reflexive only if for all 'x' in our base set we have (x, x) in the relation. Key words here being 'for all.'
A relation is irreflexive only if for all 'x' in our base set we have (x, x) is not in the relation. Key words here being 'for all.'
So the relation R1 above is not reflexive as (4, 4) and (7, 7) are missing, but R1 is also not irreflexive as (2, 2) is present.
Relation R3 is not reflexive as (2, 2), and (4, 4), and (7, 7) are missing, now because they are all missing it is now irreflexive.
R4 is an unusual case, it is the empty set, and so is vacuously reflexive and also irreflexive, it would be the only example of a set that is both.
I hope this helps.
Regards
Jonathan
Oh boy I see.
Thanks for the awesome content.
good dog
Not nice no clear voice and no clear board. And can't understand