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Question Number 212332 by ffuis2024 last updated on 10/Oct/24
  Please help    1.1.Let XUY=X for all sets X. Prove that Y=0(empty set).   From Singler book
$$ \\ $$Please help

1.1.Let XUY=X for all sets X. Prove that Y=0(empty set).
From Singler book "Excercises in set theory".
I think this task is totaly wrong and cannot be proved. I would ask someone to provide me valid proof of that. I have sets X and Y such as Y is subset of X. For example. If Y={1} and X={1,2} then XUY=X is correct but that doesn't imply Y is empty.
Another example when X=Y since X is any set. I can choose X=Y. Why not? Then YUY=Y is always true, but again, that doesnt imply Y is empty set
Proof in book claim that is correct if we suppose Y is not empty and if we choose for instance X is empty set. Then 0UY=0 but this is wrong since 0UY=Y. Therefore, Y must be empty?

Commented by A5T last updated on 10/Oct/24
The key word here is “all”. X∪Y=X must be  true for all sets X. If Y={1}, X={2,3} is   another set in all, and X∪Y≠X, so Y={1}  doesn′t satisfy.   All sets X such that X=Y is only a subset of   all. You also have to consider other sets from  all such that X≠Y.
$${The}\:{key}\:{word}\:{here}\:{is}\:“\boldsymbol{{all}}''.\:{X}\cup{Y}={X}\:{must}\:{be} \\ $$$${true}\:{for}\:\boldsymbol{{all}}\:{sets}\:{X}.\:{If}\:{Y}=\left\{\mathrm{1}\right\},\:{X}=\left\{\mathrm{2},\mathrm{3}\right\}\:{is}\: \\ $$$${another}\:{set}\:{in}\:\boldsymbol{{all}},\:{and}\:{X}\cup{Y}\neq{X},\:{so}\:{Y}=\left\{\mathrm{1}\right\} \\ $$$${doesn}'{t}\:{satisfy}.\: \\ $$$${All}\:{sets}\:{X}\:{such}\:{that}\:{X}={Y}\:{is}\:{only}\:{a}\:{subset}\:{of}\: \\ $$$$\boldsymbol{{all}}.\:{You}\:{also}\:{have}\:{to}\:{consider}\:{other}\:{sets}\:{from} \\ $$$$\boldsymbol{{all}}\:{such}\:{that}\:{X}\neq{Y}. \\ $$
Commented by ffuis2024 last updated on 10/Oct/24
I still dont get. I know that also:XUY=X⇔Y⊆X is akways true.So that means Y must be subset of X.  What means for all sets X?
$${I}\:{still}\:{dont}\:{get}.\:{I}\:{know}\:{that}\:{also}:{XUY}={X}\Leftrightarrow{Y}\subseteq{X}\:{is}\:{akways}\:{true}.{So}\:{that}\:{means}\:{Y}\:{must}\:{be}\:{subset}\:{of}\:{X}. \\ $$$${What}\:{means}\:{for}\:{all}\:{sets}\:{X}? \\ $$
Commented by A5T last updated on 10/Oct/24
A set Y which is the subset of all sets X is   empty.
$${A}\:{set}\:{Y}\:{which}\:{is}\:{the}\:{subset}\:{of}\:{all}\:{sets}\:{X}\:{is}\: \\ $$$${empty}. \\ $$
Commented by ffuis2024 last updated on 10/Oct/24
thanks a lot
$${thanks}\:{a}\:{lot} \\ $$

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