Relation Problem on A & B: What Happens to 1,3 & 2,4?

[jwxie]

This is an example from the book.

For B = {1,2}, let A = P(B) = {empty, {1}, {2}, {1,2} }
The following is an example of relation on A:

R = {
(emp,emp),
(emp, {1})
(emp, {2}),
(emp, {1,2}) ,
({1},{1}),
({1}, {1,2}) ,
({2},{2}),
({2}, {1,2},
({1,2},{1,2}),
}

My question is, what happen to ({1}, emp), ({2}, {1}.. i see the trend that it is skipping everything before the current relation. Why?

Like A X B
let A = {1,2} and B = {2,3,4}
We will have 1,2 1,3 1,4 2,2, 2,3 2,4

 

[Martin Rattigan]

A relation on a set A can be any old set of ordered pairs with first and second entries appearing in A subject only to the proviso that all of A's elements appear as first or second element of at least one ordered pair in the relation. (The proviso can also be dropped if a relation in A, rather than on A, is specified.)

 

[jwxie]

Hi, Martin, thank you. Can you explain further? Your definition is so advance for me to understand, sorry.
Thanks

 

[Martin Rattigan]

I just looked the definition up on wiki and there a relation on A is defined without the proviso I inserted (i.e. exactly as i described relation in A). This may also be the definition in your book.

 

[Martin Rattigan]

Ok if A={1,2,3} then R={(1,3),(1,1)} would be a relation on A according to wiki's definition. Here I just chose the elements of the ordered pairs (1,3) and (1,1) at random from A.

If you mean what I would normally mean by a relation on A (and what your book may mean - you'd have to check) the relation R would only be a relation in R, because 2 isn't included in any ordered pair. There is probably no distinction in general use so, unless your book says something to the contrary, you can assume that any set of ordered pairs chosen in a similar way to the way I chose R in the first paragraph will do as a binary relation on A.

Hope that makes sense.

 

[Martin Rattigan]

The fact that A=P(B) in the book's example is irrelevant by the way.

 

[jwxie]

Hi, thanks. So basically the book provides an incomplete ordered pairs of A X A relations.

 

[Martin Rattigan]

Yes. A relation in A is usually defined as a subset (any subset) of AxA.