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August 13th, 2017, 06:48 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 $\displaystyle n \in Z$, $\displaystyle (n1,n)\in S^2$ 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 08:11 AM. 
August 13th, 2017, 08:23 AM  #2  
Senior Member Joined: Oct 2009 Posts: 608 Thanks: 186  Quote:
In (1) can you not see any way to subdivide A into disjoint classes such that all are even? Quote:
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August 13th, 2017, 11:14 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 11:24 AM. 
August 14th, 2017, 12:54 PM  #4  
Newbie Joined: Aug 2017 From: Paris (France) Posts: 2 Thanks: 0 Math Focus: Probability, stochastic processes 
Hi, Did you manage to solve from (1) to (6) with the advice Micrm@ss gave you? Quote:
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August 27th, 2017, 05:25 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
An obvious counterexample 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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