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Set Theory. Cartesian product of generalised union. watch

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    • Thread Starter

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

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    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:

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


    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\in X\times Y for some X\in S,Y \in T

    And x \in X, y\in Y

    Then x\in \cup S, y\in \cup T

    So, 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.


    Can't see anything wrong with your argument.
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    • Thread Starter

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    (Original post by DFranklin)
    Can't see anything wrong with your argument.
    Thanks - PRSOM.
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