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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: 3,264 Thanks: 902 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. Tags set, theory, truthfalse Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mathsnoob Calculus 2 August 13th, 2017 03:13 AM Gab New Users 4 February 4th, 2017 11:16 AM jacksonjabbers Linear Algebra 1 October 30th, 2016 03:44 PM flextera New Users 0 July 30th, 2014 12:12 PM Meetra Applied Math 2 October 28th, 2009 07:56 AM

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