Inverse and other concepts defined for arbitrary sets.

[Barles]

So, I'm reading over Herbert Enderton's "Elements of Set Theory" and have come to the section on functions in chapter 3. There, he defines the concepts of the inverse of a set, the composition of 2 sets, the restriction of a set F to another set A, and the image of a set A under a set F. Note that I said "set" in each case, and not "function" or "relation". This is because he says that these concepts, while usually applied to functions, and sometimes to relations, can actually be defined in terms of arbitrary sets. I'll post the definitions of "inverse" and "composition" below:

(a) The inverse of a set F is the set F^-1 = {<u,v> | vFu}

(b) The composition of F and G is the set F * G = {<u,v> | There is a set t such that uGt and tFv)}

These definitions make use of sets of ordered pairs, which are by definition, at least relations if not functions. So, then, how do these definitions apply to arbitrary sets which may not be relations (sets of ordered pairs)? This is not a homework problem or anything, but not understanding this is bugging me a bit.

 

[pka]

So, I'm reading over Herbert Enderton's "Elements of Set Theory" and have come to the section on functions in chapter 3. There, he defines the concepts of the inverse of a set, the composition of 2 sets, the restriction of a set F to another set A, and the image of a set A under a set F. Note that I said "set" in each case, and not "function" or "relation". This is because he says that these concepts, while usually applied to functions, and sometimes to relations, can actually be defined in terms of arbitrary sets. I'll post the definitions of "inverse" and "composition" below:

(a) The inverse of a set F is the set F^-1 = {<u,v> | vFu}

(b) The composition of F and G is the set F * G = {<u,v> | There is a set t such that uGt and tFv)}

These definitions make use of sets of ordered pairs, which are by definition, at least relations if not functions. So, then, how do these definitions apply to arbitrary sets which may not be relations (setsatical of ordered pairs)? This is not a homework problem or anything, but not understanding this is bugging me a bit.

Here is Herbert Enderton's mathematical genealogy. He was a student of Putnam who was a philosopher(logician). Hence his writings are a bit different from the typical mathematics textbook. I have just look over the material you have asked about. If you notice the title of chapter 3 is Relations and Functions I think you are reading too much into his definitions. Both a relation and a function are sets of ordered pairs. So any remarks he makes about in a) or b) above apply to ordered pairs. It would be a mistake to think the they can be applied to sets in general.

 

[Barles]

I feel like if I just leave it at that, I'm copping out. However, leaving it at that doesn't stop me from proceeding with the rest of the material in book (at least as far ahead as I've peeked), so I might as well. Thanks for the advice.

 

[HallsofIvy]

So, I'm reading over Herbert Enderton's "Elements of Set Theory" and have come to the section on functions in chapter 3. There, he defines the concepts of the inverse of a set, the composition of 2 sets, the restriction of a set F to another set A, and the image of a set A under a set F. Note that I said "set" in each case, and not "function" or "relation". This is because he says that these concepts, while usually applied to functions, and sometimes to relations, can actually be defined in terms of arbitrary sets. I'll post the definitions of "inverse" and "composition" below:

(a) The inverse of a set F is the set F^-1 = {<u,v> | vFu}

(b) The composition of F and G is the set F * G = {<u,v> | There is a set t such that uGt and tFv)}

These definitions make use of sets of ordered pairs, which are by definition, at least relations if not functions. So, then, how do these definitions apply to arbitrary sets which may not be relations (sets of ordered pairs)? This is not a homework problem or anything, but not understanding this is bugging me a bit.

For either of these to make sense, you will need to say what "vFu" means for set F How does Enderton define that?

 

[pka]

For either of these to make sense, you will need to say what "vFu" means for set F How does Enderton define that?

Enderton clearly defines it in the chapter on Relations and Functions.
\(\displaystyle F\) is a relation or function and \(\displaystyle vFu\) and it simply means \(\displaystyle <u,v>\in F \).
\(\displaystyle <u,v> \) is Enderton's notation for ordered pair. As I said above, his logic training was in philosophy at Harvard under Hillary Putnam.

 

[HallsofIvy]

Yes, that's pretty standard definition for a relation as well as for the "inverse" of a relation but the OP confused me by saying "he defines the concepts of the inverse of a set", not a relation.

 

[Barles]

Yes, that's pretty standard definition for a relation as well as for the "inverse" of a relation but the OP confused me by saying "he defines the concepts of the inverse of a set", not a relation.

Sorry for the confusion. I said that because right before giving the definitions, he makes this statement:

The following operations are most commonly applied to functions, sometimes are applied to relations, but can actually be defined for arbitrary sets A, F,and G.

He then gives the definitions, two of which are in the opening post. That left *me* in confusion.