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September 16th, 2018, 04:59 AM   #1
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algebraic structures

A binary relation is a set formed by pairs taken from the Cartesian product between two sets, according to a "rule" that varies from relation to relation. In particular, let us consider, in the set of positive integers, the binary relation * defined by a * b = c, where c is the greatest divisor common between a and b. Class V for true sentences and F for false sentences:

() * is commutative.
() * is associative.
() 1 is the neutral element.
() a * a = a, for all a.
() For each a, there exists b such that a * b = 1.

Now, check the alternative that shows the sequence CORRECT:
* a) F-V-F-F-F.
* b) V-F-V-V-V.
* c) F-V-F-F-V.
* d) V-V-V-V-F.

Can someone help please?
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September 16th, 2018, 06:35 AM   #2
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It's not your fault, but this question is awful. The definition it gives of a binary relation is very imprecise and is more complicated than is needed: a binary relation on a set $X$ is a subset $R$ of the cartesian product $X \times X$. That's it. While most binary relations we're interested in might be given by some "rule" (knowing that $(a,b) \in R$ if and only if $a$ and $b$ satisfy some nicely formulated property), this is certainly not necessary.

Furthermore, the example it gives is of a binary operation, and not a binary relation. Note that it doesn't really make sense to talk about commutativity or associativity of a binary relation (the closest analogue to commutativity is symmetry, which says $(a,b) \in R$ if and only if $(b,a) \in R$, and there isn't an analogue of associativity). A binary operation on $X$ is a function $f: X \times X \to X$, where we often write $a * b$ (or even just $ab$) as shorthand for $f(a,b)$. Here we have $X = \mathbb{Z}_{>0}$ and $a * b = \operatorname{gcd}(a,b)$.

Now to get to the actual problem:

1) This is simply asking whether $\operatorname{gcd}(a,b) = \operatorname{gcd}(b,a)$ for all positive integers $a$ and $b$. It should be clear that this is true.

2) This is asking whether $\operatorname{gcd}(a, \operatorname{gcd}(b,c)) = \operatorname{gcd}(\operatorname{gcd}(a,b), c)$ for all positive integers $a, b$ and $c$. This isn't hard to show directly from the definition of $\operatorname{gcd}$; alternatively you could use the fundamental theorem of arithmetic and work with prime decompositions of $a,b$ and $c$ to get it.

3) Given that the first two parts are true, we are locked into option d) which means this is also true. However, this is a little odd: when we say $e$ is a "neutral" or "identity" element, we usually mean that it leaves other elements unchanged: $a * e = e * a = a$ always. But here $1$ kills off all other elements: $a * 1 = \operatorname{gcd}(a,1) = 1$ for all positive integers $a$. I guess here the word "neutral" is being used in the sense of "neutralizing" or "neutering". (That, or the question is wrong.)

4) This says $\operatorname{gcd}(a,a) = a$. This one should also be clear to you.

5) This says for every positive integer, there is a positive integer coprime to it. This is true (hint: see my discussion for part 3)).

So really we have V-V-V/F(depending on what "neutral" actually means here)-V-V, which is option e): none of the above.
Thanks from Roberto 37

Last edited by cjem; September 16th, 2018 at 06:42 AM.
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September 16th, 2018, 07:04 AM   #3
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Can a photograph of the original question be posted, so that we can check for typing or translation errors?
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September 16th, 2018, 02:08 PM   #4
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Sorry, it's in Portuguese.
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September 16th, 2018, 02:11 PM   #5
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September 16th, 2018, 02:35 PM   #6
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V - V - V - V - F is it corret? do you think?
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September 16th, 2018, 03:05 PM   #7
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Quote:
Originally Posted by Roberto 37 View Post
V - V - V - V - F is it corret? do you think?
As I said, the answer should be V - V - F - V - V (or V - V - V - V - V if "neutral"/"neutro" is being meant in a weird way). All of the given options a), b), c) and d) are incorrect.
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September 16th, 2018, 05:42 PM   #8
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but is for options below, only 1.

* a) F-V-F-F-F.
* b) V-F-V-V-V.
* c) F-V-F-F-V.
* d) V-V-V-V-F.

I think letter d.

Last edited by Roberto 37; September 16th, 2018 at 05:46 PM.
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September 17th, 2018, 09:17 AM   #9
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Quote:
Originally Posted by Roberto 37 View Post
but is for options below, only 1.

* a) F-V-F-F-F.
* b) V-F-V-V-V.
* c) F-V-F-F-V.
* d) V-V-V-V-F.

I think letter d.
None of the options are correct.
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September 17th, 2018, 02:22 PM   #10
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but I need to check 1
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