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setcartesian-productpowersetset-theory
aniketzr
aniketzr

Reputation: 73

cartesian product of a power set with a set

how to explicitly write the cartesian product of a power set with another set.

eg: P({a,b})x{a,b}

Now P({a,b}) = {{},{a},{b},{a,b}}

so i need to know {{},{a},{b},{a,b}}x{a,b}

Upvotes: 2

Views: 8180

Answers (2)

Nate Chandler
Nate Chandler

Reputation: 4553

Let X be a a set. The power set of X is defined to be

P(X) := { S | S ⊆ X }

Let X and Y be sets. The product X × Y is defined to be

X × Y := { (x,y) | x ∈ X, y ∈ Y }

Now let X and Y be sets. We will describe the Cartesian product of the power set of X with Y:

P(X) × Y = { (S,y) | S ∈ P(X), y ∈ Y }

But S ∈ P(X) if and only if S ⊆ X. This allows us to rewrite our product as

P(X) × Y = { (S,y) | S ⊆ X, y ∈ Y }

In other words, P(X) × Y consists of ordered pairs such that the first coordinate is some subset of X and the second coordinate is an element of Y.

Upvotes: 2

Edouard Berthe
Edouard Berthe

Reputation: 1463

  1. Your example is not consistent with your question:

You ask "how to explicitly write the cartesian product of a power set with another set?", and then you give the example of P({a,b})x{a,b}, which is the cartesian product of a power set with the same set, namely {a,b}.

  1. For your example, as you said you have:

P({a,b}) = {{},{a},{b},{a,b}}

Then recall the definition of the cartesian product of two sets:

E x F = {(e,f), e in E, f in F}

Then by applying this definition:

P({a,b}) x {a,b} = {{},{a},{b},{a,b}} x {a,b} = {({}, a), ({}, b), ({a}, a), ({a}, b), ({b}, a), ({b}, b), ({a,b}, a), ({a,b}, b)}

However, it is very unlikely that this kind of property would be useful in any concrete case or even theoretical theorem, because we almost never meet cases where {a} and a are treated "equally" (I mean, on the same "level" of sets), although it is absolutely correct from a mathematical point of view.

Upvotes: 0

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