# Thread: Relation From Set A to B

1. ## 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...

2. ## Re: Relation From Set A to B

Hello, Bashyboy!

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

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: . $\text{gcd}(0,0) \,=\,\text{any number}\ne 0$

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: . $\begin{Bmatrix}\text{lcm}(1,2)\,=\,2 \\ \text{lcm}(2,1)\,=\,2 \end{Bmatrix}$

3. ## Re: Relation From Set A to B

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.

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.

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.

4. ## 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.

5. ## Re: Relation From Set A to B

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).