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?


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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 "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

$T h e k e y w o r d h e r e i s ‘ ‘ {a l l}^{″} . X \cup Y = X m u s t b e$ $t r u e f o r a l l s e t s X . I f Y = {1}, X = {2, 3} i s$ $a n o t h e r s e t i n a l l, a n d X \cup Y \neq X, s o Y = {1}$ ${d o e s n}^{'} t s a t i s f y .$ $A l l s e t s X s u c h t h a t X = Y i s o n l y a s u b s e t o f$ $a l l . Y o u a l s o h a v e t o c o n s i d e r o t h e r s e t s f r o m$ $a l l s u c h t h a t X \neq Y .$
Commented by ffuis2024 last updated on 10/Oct/24

$I s t i l l d o n t g e t . I k n o w t h a t a l s o : X U Y = X \Leftrightarrow Y \subseteq X i s a k w a y s t r u e . S o t h a t m e a n s Y m u s t b e s u b s e t o f X .$ $W h a t m e a n s f o r a l l s e t s X ?$
Commented by A5T last updated on 10/Oct/24

$A s e t Y w h i c h i s t h e s u b s e t o f a l l s e t s X i s$ $e m p t y .$
Commented by ffuis2024 last updated on 10/Oct/24

$t h a n k s a l o t$
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