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If R = {(1, 1), (2, 2), (1, 2), (2, 1), (3, 3)} and S = {(1, 1), (2, 2), (2, 3), (3, 2),
(3, 3)} are two relations in the set X = {1, 2, 3}, the incorrect statement is:
(A) R and S are both equivalence relations
(B) R∩S is an equivalence relations
(C)R^(-1)∩S^(-1) is an equivalence relations
(D) R∪ S is an equivalence relations
With Regarks,
Prakash Panneer
Last edited by Prakash Panneer (2006-06-06 02:30:21)
Letter, number, arts and science
of living kinds, both are the eyes.
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I take it these are permutations and not ordered pairs. For example, R goes from 1 to 1, and 1 to 2, and so on.
To check if they are equivalence relations, you need to check:
a. Every number is related to itself
b. If a is related to b, then b is related to a.
c. If a is related to b, and b is related to c, then a is related to c.
For R and S, these are fairly easy to check.
R∩S = {(1, 1), (2, 2), (3, 3)} which is an equivalence relation
R^(-1)∩S^(-1)
R^(-1) = {(1, 1), (2, 2), (2, 1), (1, 2), (3, 3)}
S^(-1) = {(1, 1), (2, 2), (3, 2), (2, 3), (3, 3)}
Which should look familar.
So that means we are left with D. Determine why D can't be an equivalence relation.
"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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:-) Many thanks for your help Ricky :-)
Letter, number, arts and science
of living kinds, both are the eyes.
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