Re: Relation From Set A to B

Hello, Bashyboy!

Quote:

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

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

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: .$\displaystyle \text{gcd}(0,17) \,=\,17$

What is the GCD of 0 and 0?

Both 0 and 0 are divisible by $\displaystyle \text{ any number}\ne 0$

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

Quote:

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

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

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: .$\displaystyle \text{lcm}(4,2) = 4$

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

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 $\displaystyle \text{ any number}\ne 0$

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

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

Quote:

Originally Posted by

**Bashyboy** $\displaystyle A = \{0, 1, 2, 3, 4\}\and\B = \{0, 1, 2, 3\},a\epsilon A \and\ b \epsilon B$

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.

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.

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).