# Relation From Set A to B

• October 30th 2012, 02:52 PM
Bashyboy
Relation From Set A to B
$A = \{0, 1, 2, 3, 4\}\and\B = \{0, 1, 2, 3\},a\epsilon A \and\ b \epsilon B$

I need to find the Relation such that $R = \{(a, b)|\gcd(a, b)=1\}$

I included (0, 0) as one of my ordered pairs in the relation set, but apparently it isn't suppose to be in it. Why is that? Isn't 0 divisible by one?

And for the same two sets, A and B, I have to find the relation $R=\{(a,b)|lcm(a, b)=2\}$

What I found was $R=\{(2,2),(4,2)\}$

But the only correct ordered-pair in that set is (2, 2). I'm not entirely sure what I did wrong...
• October 30th 2012, 05:09 PM
Soroban
Re: Relation From Set A to B
Hello, Bashyboy!

Quote:

$A = \{0, 1, 2, 3, 4\}\text{ and }B = \{0, 1, 2, 3\},\;a\in A\text{ and }b \in B$

I need to find the Relation such that: $R \:=\: \{(a, b)\,|\,\gcd(a, b)=1\}$

I included (0, 0) as one of my ordered pairs in the relation set, but apparently it isn't suppose to be in it.
Why is that? .Isn't 0 divisible by one?

By convention, we do not consider zero in discussions of GCDs and LCMs.

Recall the defintion of a GCD.
. . It is the greatest number that divides into two (or more) numbers.

What is the GCD of 0 and 17?
Both 0 and 17 are divisible by 17.
. . Hence: . $\text{gcd}(0,17) \,=\,17$

What is the GCD of 0 and 0?
Both 0 and 0 are divisible by $\text{ any number}\ne 0$
. . Hence: . $\text{gcd}(0,0) \,=\,\text{any number}\ne 0$

Quote:

And for the same two sets, $A$ and $B$, I have to find the relation: $R\:=\:\{(a,b)\,|\,\text{lcm}(a, b)=2\}$

What I found was $R=\{(2,2),(4,2)\}$

But the only correct ordered-pair in that set is (2, 2).
I'm not entirely sure what I did wrong.

Your second pair is incorrect: . $\text{lcm}(4,2) = 4$

I would include: . $\begin{Bmatrix}\text{lcm}(1,2)\,=\,2 \\ \text{lcm}(2,1)\,=\,2 \end{Bmatrix}$
• November 2nd 2012, 05:10 AM
emakarov
Re: Relation From Set A to B
Quote:

Originally Posted by Bashyboy
{(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}

According to the answer key, it has none of the properties. I was thinking it was perhaps antisymmetric and transitive. Why isn't it so?

(1, 3) and (3, 1) are in the relation. This pertains to both transitivity and antisymmetry.

Quote:

Originally Posted by Soroban
What is the GCD of 0 and 0?
Both 0 and 0 are divisible by $\text{ any number}\ne 0$
. . Hence: . $\text{gcd}(0,0) \,=\,\text{any number}\ne 0$

Rather, 0 and 0 do not have the greatest common divisor because any number is a divisor.

Quote:

Originally Posted by Bashyboy
$A = \{0, 1, 2, 3, 4\}\and\B = \{0, 1, 2, 3\},a\epsilon A \and\ b \epsilon B$

A stylistic remark: In this case, instead of "a ∈ A and b ∈ B," it should say "a ranges over A and b ranges over B." The former phrase raises questions whether you are talking about some specific a and b that you forgot to introduce or you are considering some arbitrary a and b and, if so, for what reason. The latter phrase means that whenever a is used in what follows, the reader may assume that it is some element of A.
• November 2nd 2012, 05:20 AM
Bashyboy
Re: Relation From Set A to B
So, are you saying that, perhaps, the answer key is wrong; and that {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)} is transitive and antisymmetric?

Also, thank you for your last remark, concerning the phrasing of certain things.
• November 2nd 2012, 06:55 AM
emakarov
Re: Relation From Set A to B
Quote:

Originally Posted by Bashyboy
So, are you saying that, perhaps, the answer key is wrong; and that {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)} is transitive and antisymmetric?

No, the answer key is correct, and you can build counterexamples using (1, 3) and (3, 1).