September 16th, 2018, 03:59 AM  #1 
Newbie Joined: Jun 2018 From: Brasil Posts: 13 Thanks: 0  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) FVFFF. * b) VFVVV. * c) FVFFV. * d) VVVVF. Can someone help please? 
September 16th, 2018, 05:35 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 219 Thanks: 70 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
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 VVV/F(depending on what "neutral" actually means here)VV, which is option e): none of the above. Last edited by cjem; September 16th, 2018 at 05:42 AM. 
September 16th, 2018, 06:04 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,509 Thanks: 1741 
Can a photograph of the original question be posted, so that we can check for typing or translation errors?

September 16th, 2018, 01:08 PM  #4 
Newbie Joined: Jun 2018 From: Brasil Posts: 13 Thanks: 0 
Sorry, it's in Portuguese.

September 16th, 2018, 01:11 PM  #5 
Newbie Joined: Jun 2018 From: Brasil Posts: 13 Thanks: 0  photo
phto

September 16th, 2018, 01:35 PM  #6 
Newbie Joined: Jun 2018 From: Brasil Posts: 13 Thanks: 0 
V  V  V  V  F is it corret? do you think?

September 16th, 2018, 02:05 PM  #7 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 219 Thanks: 70 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
September 16th, 2018, 04:42 PM  #8 
Newbie Joined: Jun 2018 From: Brasil Posts: 13 Thanks: 0 
but is for options below, only 1. * a) FVFFF. * b) VFVVV. * c) FVFFV. * d) VVVVF. I think letter d. Last edited by Roberto 37; September 16th, 2018 at 04:46 PM. 
September 17th, 2018, 08:17 AM  #9 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 219 Thanks: 70 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
September 17th, 2018, 01:22 PM  #10 
Newbie Joined: Jun 2018 From: Brasil Posts: 13 Thanks: 0 
but I need to check 1


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