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Set Theory. Cartesian product of generalised union.

This is a question taken from "The Foundations of Mathematics" by Stewart and Tall (2nd edition).

Untitled.jpg

My query is with the statement "Show that in the first formula we cannot replace '⊆\subseteq' with '=='. I can see no justification for it not being equality.

I've checked both editions of the book, and that phrase is in each, though they have changed the notation slightly - which makes me think they have looked at the question and think that statement should stand. First formula, in the 1st edition, used to read:

(∪S)×(∪T)⊆∪(S×T)(\cup S)\times (\cup T) \subseteq \cup (S\times T)
Edit: Corrected above line.

Working

Looking at that first formula, just the part to show RHS is a subset of LHS.

Let z=(x,y) be an element of the RHS.

Then z∈X×Yz\in X\times Y for some X∈S,Y∈TX\in S,Y \in T

And x∈X,y∈Yx \in X, y\in Y

Then x∈∪S,y∈∪Tx\in \cup S, y\in \cup T

So, z=(x,y)∈(∪S)×(∪T)z=(x,y)\in (\cup S)\times (\cup T) and is an element of the LHS.

Am I missing some subtlety there, or is the book in error. Any thoughts appreciated.

G.
(edited 7 years ago)
Can't see anything wrong with your argument.
Original post by DFranklin
Can't see anything wrong with your argument.


Thanks - PRSOM.

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