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Question

Power Set and Union

Answer 1

+2 A:

you have

M := Pow(X) u {Y}

with

Pow(X) := {0, {Horse}, {Dog}, {Horse, Dog}}

so

M := {0, {Horse}, {Dog}, {Horse, Dog}} u {{Cat}}

Does that clear it up for you?

The set you've displayed the union mapped over the cartesian product and missing {Cat}.

aaronasterling 2010-10-21 05:12:41

Hm, but cat has to be added to the other elements since it is a union!? like M := {{Horse, Cat}, {Dog, Cat}, {Horse, Dog, Cat}} or is that wrong?

ArtWorkAD 2010-10-21 05:15:29

@ArtWordAD, It's the union of the set Pow(X) with {Cat}. It's Not the union of {Cat} with each individual element of Pow(X) which is what you are trying to do. Do you see the distinction?

aaronasterling 2010-10-21 05:19:01

so what would be the resulting set of M := {0, {Horse}, {Dog}, {Horse, Dog}} u {Cat}?

ArtWorkAD 2010-10-21 05:21:09

@ArtWorkAD, I think it would help you to replace Pow(X) with elements that are not sets and take the union of that set with {Cat} to see how it works. You can then substitute the elements of the replacement set for the corresponding elements of Pow(X) in the union and you will have your answer. Another alternative would be to review the definition of the union of two sets and look for a corner case in it where one of the sets contains sets. [hint: you won't find it]

aaronasterling 2010-10-21 05:24:15

thanks, so M := {0, {Horse}, {Dog}, {Horse, Dog}, Cat} would be the final set?

ArtWorkAD 2010-10-21 05:27:35

@ArtWordAD Looks good to me.

aaronasterling 2010-10-21 05:32:37

{Y} = {{Cat}} not {Cat}.

andand 2010-10-21 14:49:07

@andland good eye. I missed the substitution.

aaronasterling 2010-10-21 20:13:53

@ArtWorkAD, we both made a mistake but andand caught it.

aaronasterling 2010-10-21 20:50:34

Answer 2

+1 A:

M := {0, {Horse}, {Dog}, {Horse, Dog}, Cat}

The definition of the union is the set of elements that are in either set. So {Horse,Cat} is not in the union, because it is not in either set.

Beta 2010-10-21 05:21:41

Answer 3

+3 A:

I'm going to differ slightly with the other responses. If you define Y = {Cat} then {Y} = {{Cat}}, that is, Y is the set containing the element Cat and {Y} is the set containing Y, or the set containing the set containing the element Cat. In that case:

M := {0, {Horse}, {Dog}, {Horse, Dog}, {Cat} }

It's a subtle, but important distinction in set theory.

andand 2010-10-21 14:47:20

There's not actually much subtlety about it

aaronasterling 2010-10-21 20:15:31

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