Binary relations: reflexive, symmetric, antisymmetric, transitive

Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive.

The relation R on the set {(m, n)|(m, n) ∈ Z}, where (m, n)R(x,y) when m = x and n = y.

This is unlike the problems I've usually seen and I'm not sure how to answer it.

Any direction would be appreciated.

Re: Binary relations: reflexive, symmetric, antisymmetric, transitive

Quote:

Originally Posted by

**nicnicman** Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive.

The relation R on the set {(m, n)|(m, n) ∈ Z}, where (m, n)R(x,y) when m = x and n = y.

Sorry to say, but that is nonsense.

Read what you posted again.

Do you mean $\displaystyle \{(m,n): (m,n)\in\mathbb{Z}\times\mathbb{Z}\}~?$

What you wrote is nonsense.

Then if it is $\displaystyle (m,n)R(x,y)$ if $\displaystyle m=x~\&~n=y$ that more nonsense because that is simply equality.

Please correct the question.

Re: Binary relations: reflexive, symmetric, antisymmetric, transitive

That's exactly how the question was given.

For the first part, {(m, n)|(m, n) ∈ Z}, I assume means (m,n) such that (m,n) is a member of the set of integers. Does that make sense?

I guess if the question doesn't make sense, none of the relationships exist.

Re: Binary relations: reflexive, symmetric, antisymmetric, transitive

Quote:

Originally Posted by

**nicnicman** That's exactly how the question was given.

For the first part, {(m, n)|(m, n) ∈ Z}, I assume means (m,n) such that (m,n) is a member of the set of integers. Does that make sense?

Whoever wrote that question does not know enough to be teaching that course.

Re: Binary relations: reflexive, symmetric, antisymmetric, transitive

Could be some sort of typo. Guess I'll just write that the question doesn't make sense so none of the relationships exist.

Thanks for your help.

Re: Binary relations: reflexive, symmetric, antisymmetric, transitive

Quote:

Originally Posted by

**nicnicman** The relation R on the set {(m, n)|(m, n) ∈ Z}, where (m, n)R(x,y) when m = x and n = y.

The problem is that (m, n) ∈ ℤ is a type mismatch. Here (m, n) is a pair of integers; they can only belong to ℤ × ℤ. If you want to say that m and n are integers, you can write "m, n ∈ ℤ," which is an abbreviation for "m ∈ ℤ and n ∈ ℤ."

Also, as Plato noted, (m, n) R (x,y) iff m = x and n = y makes R a simple equality on pairs (two pairs are equal if their corresponding elements are equal). It's trivial, but it can be checked that equality is an equivalence relation. Are you sure it's not "(m, n)R(x,y) when m = x **or** n = y"?

Concerning "This is unlike the problems I've usually seen," you should first look at several examples of proving that a relation is reflexive, symmetric or transitive. Find them in your textbook or lecture notes or search this forum. This is an absolutely standard type of problem.

Re: Binary relations: reflexive, symmetric, antisymmetric, transitive

Quote:

Concerning "This is unlike the problems I've usually seen," you should first look at several examples of proving that a relation is reflexive, symmetric or transitive. Find them in your textbook or lecture notes or search this forum. This is an absolutely standard type of problem.

Yeah, we've done a bunch of these problems. What I meant is that this is unlike the problems of this type that I've seen. Thus, I have no idea how to answer it. We're only given reflexive, symmetric, antisymmetric and transitive as relationship options. Equivalence is not given as an option.

I checked again and this is exactly as the problem is written.

Thanks for your help.