Power Set and Union

Following set is given:

X := {Horse, Dog} 
Y := {Cat}

I d开发者_StackOverflow中文版efine the set:

M := Pow(X) u {Y}

u for union

The resulting set of the power set operation is:

Px := {0, {Horse}, {Dog}, {Horse, Dog}}

0 for empty set

My question is referenced to the unio operation. How do I unite 0 and Y?

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

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.

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}.

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.