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 August 13th, 2017, 05:48 AM #1 Newbie   Joined: Aug 2017 From: mathmatistan Posts: 9 Thanks: 0 several truth\false from set theory Apparently, I've solved the wrong set of exercises and now I have to solve a different bunch (lol). the questions are true/false types. I'll write what I did/tried or got wrong for every question: 1) for every equivalence relation over A={1,2,3,4,5,6} there is a class in which the number of variables is odd. Even though the number of variables is not odd, there could be an odd class with odd number of variables. but I don't know whether it's suffice to assert it's a true claim. 2) for every equivalence relation over A={1,2,3,4,5,6,7} there is a class in which the number of variables is odd. Seems true, since the number of variables inside A is odd. 3) if R is symmetrical and reflexive over A={1,2} then R is equivalence relation. To be called an equivalence relation, it should hold: reflexive relation, transitive relation and and symmetric relation. in this case, I think it's a true claim. 4) if R is symmetrical and reflexive over A={1,2,3} then R is equivalence relation. I think that in this case, it is not not a true claim because of the odd number of variables. 5) if $\displaystyle R^2$ is symmetrical and reflexive over A={1,2,3}, then R is equivalence relation. R^2 is defined as R x R, but I still don't know whether it's a true claim or not. I think that not. if I'm wrong, please show me why. 6) if S is equivalence relation, then every one of its classes has the same number of variables. Logically, it seems to be true. I don't see a reason why this sentence would be wrong. 7) there exist even number $\displaystyle m,n \in Z$ so that $\displaystyle (n,m) \in S$ Seems to be true. Again, I don't see any reason why it would be false or example that would contradict it. 8) for every $\displaystyle n \in Z$, $\displaystyle (-n-1,n)\in S^2$ Since we're talking about $\displaystyle S^2$, then negative numbers become positive, and thus it seems logical that this sentence will be true. I've tried to give detailed reasoning for my questions so you could understand why I chose a certain answer. Please correct me if I'm wrong and help me improve and become better. Last edited by skipjack; August 13th, 2017 at 07:11 AM.
August 13th, 2017, 07:23 AM   #2
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Quote:
 Originally Posted by mathsnoob Apparently, I've solved the wrong set of exercises and now I have to solve a different bunch (lol). the questions are true/false types. I'll write what I did/tried or got wrong for every question: 1) for every equivalence relation over A={1,2,3,4,5,6} there is a class in which the number of variables is odd. Even though the number of variables is not odd, there could be an odd class with odd number of variables. but I don't know whether it's suffice to assert it's a true claim. 2) for every equivalence relation over A={1,2,3,4,5,6,7} there is a class in which the number of variables is odd. Seems true, since the number of variables inside A is odd.
The point is whether it is true for ANY equivalence relation. So if I subdivides the A in (2) in {1, 2, 3}, {4, 5}, {6, 7}, then it's true for this one. But is it true for any one?

In (1) can you not see any way to subdivide A into disjoint classes such that all are even?

Quote:
 3) if R is symmetrical and reflexive over A={1,2} then R is equivalence relation. To be called an equivalence relation, it should hold: reflexive relation, transitive relation and and symmetric relation. in this case, I think it's a true claim.
You need to check whether a~b and b~c implies a~c. You can easy do this by brute force. Find all symmetric and reflexive relations on A. Check whether they are transitive. There aren't a lot of them.

Quote:
 4) if R is symmetrical and reflexive over A={1,2,3} then R is equivalence relation. I think that in this case, it is not not a true claim because of the odd number of variables.
It has nothing to do with odd and even. Can you find a symmetric and reflexive relation that is not transitive? There aren't a lot.

Quote:
 5) if $\displaystyle R^2$ is symmetrical and reflexive over A={1,2,3}, then R is equivalence relation. R^2 is defined as R x R, but I still don't know whether it's a true claim or not. I think that not. if I'm wrong, please show me why.
What are the possibilities for R?

Quote:
 6) if S is equivalence relation, then every one of its classes has the same number of variables. Logically, it seems to be true. I don't see a reason why this sentence would be wrong.
What about {1,2}, {3} on A={1,2,3}?

Quote:
 7) there exist even number $\displaystyle m,n \in Z$ so that $\displaystyle (n,m) \in S$ Seems to be true. Again, I don't see any reason why it would be false or example that would contradict it.
What is Z, what is S?

Quote:
 8) for every $\displaystyle n \in Z$, $\displaystyle (-n-1,n)\in S^2$ Since we're talking about $\displaystyle S^2$, then negative numbers become positive, and thus it seems logical that this sentence will be true.
What is Z? What is S?

 August 13th, 2017, 10:14 AM #3 Newbie   Joined: Aug 2017 From: mathmatistan Posts: 9 Thanks: 0 Sorry for not providing the full information regarding questions 6,7,8: S is a relation on the integer set(Z) that applies $\displaystyle (n,m) \in Z \text{ iff } m^2+m=n^2+n$ Last edited by skipjack; August 13th, 2017 at 10:24 AM.
August 14th, 2017, 11:54 AM   #4
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Hi,

Did you manage to solve from (1) to (6) with the advice Micrm@ss gave you?

Quote:
 Originally Posted by mathsnoob S is a relation on the integer set(Z) that applies $\displaystyle (n,m) \in Z \text{ iff } m^2+m=n^2+n$ 7) there exist even number $\displaystyle m,n \in Z$ so that $\displaystyle (n,m) \in S$ Seems to be true. Again, I don't see any reason why it would be false or example that would contradict it.
Don't you think $\displaystyle S$ is reflexive? Then choose any even number, it will be in relation with itself.

Quote:
 Originally Posted by mathsnoob 8) for every $\displaystyle n \in Z$, $\displaystyle (-n-1,n)\in S^2$ Since we're talking about $\displaystyle S^2$, then negative numbers become positive, and thus it seems logical that this sentence will be true.
You're just asked whether for all $\displaystyle n\in Z$ you have $\displaystyle (-n-1)^2+(-n-1)=n^2+n$

 August 27th, 2017, 04:25 AM #5 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866 An obvious counter-example for (1) is the relation in which every member of A is related to all others. The only equivalence classes are A itself and the empty set. They contain an even number of elements.

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