I don't understand how to do this problem if anyone could help that would be great
Determine which of the following define equivalence relations in R^2. If the relation fails to be an equiv- alence relation, identify at least one property that does not hold and give an example to demonstrate. If the relation is indeed an equivalence relation, give a geometrical interpretation of the quotient set.
(a.) (a,b) ~(c,d) if and only if ab = cd
(b.) (a,b)~(c,d)if and only if 2a + 3b = 2c + 3d
(c.) (a,b) ~(c,d) if and only if a + b^2 = d+c^2
When you ask a question like that people are going to wonder..
Do you know what an equivalence relation is?
Assuming that you know the definition, have you tried to check any of them against the definition?
How far did you get?
Or someone might just come along and give you the answer.
this is what I got i'm kinda stuck with the last one exponents get me worried lol.
a.) (a,b) ~(c,d) if and only if ab = cd
Reflexive: Since ab = cd then you can replace cd with ab and it would be ab = ab. So this relation is reflexive.
Symmetric: Since ab = cd then cd = ab. So then this relation is equal.
Transitive: Since ab = cd in order for this relation to be tranistive cd = ef. Which cd = ef so ab = cd and cd = ef so this relation is tranistive.
Since this relation is Reflexive, Symmetric, and Transitve then this relation is an equivalence relation.
(b.) (a,b)~(c,d)if and only if 2a + 3b = 2c + 3d
Reflexive: Since 2a + 3b = 2c + 3d then if we replace 2c + 3d with 2a + 3b, then 2a + 3b = 2a + 3b. Then this makes the relation reflexive.
Symmetric: Since 2a + 3b = 2c + 3d then 2c + 3d = 2a + 3b. Since 2c + 3d = 2a + 3b then this relation is symmetric.
Transitive: If 2a + 3b = 2c + 3d and 2c + 3d = 2e + 3f then 2a + 3b = 2e + 3f. So this relation is Transitive.
Since this relation is reflexive, symmetric, and transitive then this reltion is an equivalence relation.
That's better.Originally Posted by ajr523
Any thoughts on the geometrical interpretations?
What do you think about c)?
What happened when you checked to see if it's symmetric? Can you give and example that shows that it's not symmetric?
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2Thanks
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