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What is a relation? My book says

If A is a set, then R[subset] A x A is called a relation on A. Does this mean R[subset]A

Then the book talks about ordering. And for partial orderings it uses the notation a[<=]b rather than a~b. Does this mean less than or equal to, or is it just another notation to mean related to? If it is the former, it seems silly to me that a linear ordering is a partial ordering with the additional property that if (a,b)E R, then a[<=]b or b[<=]a because one of those two statements must be true (if not both)!

Then the book says A=R with ordinary ordering is a linear ordering. This is nonsense, because we're talking about relations. Wouldn't we have to say A[subset]R^2 before we can talk about ordering?

As an example, the book gives, Define a relation on A=Z by n~m iff n-m=3k where kEZ. All I can say is if R[subset]Z^2, R can be defined by all points (n,m) such that mEZ and n=3k +m, where kEZ, but that seems rather simplistic, and I used no math to show that result.

Someone please help me!

If A is a set, then R[subset] A x A is called a relation on A. Does this mean R[subset]A

^{2}or R, a subset of A, multiplied by A?Then the book talks about ordering. And for partial orderings it uses the notation a[<=]b rather than a~b. Does this mean less than or equal to, or is it just another notation to mean related to? If it is the former, it seems silly to me that a linear ordering is a partial ordering with the additional property that if (a,b)E R, then a[<=]b or b[<=]a because one of those two statements must be true (if not both)!

Then the book says A=R with ordinary ordering is a linear ordering. This is nonsense, because we're talking about relations. Wouldn't we have to say A[subset]R^2 before we can talk about ordering?

As an example, the book gives, Define a relation on A=Z by n~m iff n-m=3k where kEZ. All I can say is if R[subset]Z^2, R can be defined by all points (n,m) such that mEZ and n=3k +m, where kEZ, but that seems rather simplistic, and I used no math to show that result.

Someone please help me!

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