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Hard Time Understanding If A C B, then A U B = B

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Hard Time Understanding "If A C B, then A U B = B"

Homework Statement



Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.


Homework Equations



None.


The Attempt at a Solution



Since A U B = B, it can be separated into two cases. That is,

1) A U B [tex]\subseteq[/tex] B
2) B [tex]\subseteq[/tex] A U B

For case (1), I let x [tex]\in[/tex] A U B. Thus, either x [tex]\in[/tex] A or x [tex]\in[/tex] B. If x [tex]\in[/tex] A, then x [tex]\in[/tex] B. This means that A U B [tex]\subseteq[/tex] B.

For case (2), I let x [tex]\in[/tex] B. This is where I got stuck... I know I am supposed to apply the assumption that A [tex]\supseteq[/tex] B, but I am starting to think that it is impossible.

Here is an attachment of why I think it is impossible. Here is the link to tinypic for those who are too afraid to download attachments: http://tinypic.com/view.php?pic=xszle&s=7.

Can anyone give me a tip on how to approach this problem?
 

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Answers and Replies

  • #2
LCKurtz
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Homework Statement



Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.


Homework Equations



None.


The Attempt at a Solution



Since A U B = B, it can be separated into two cases. That is,

1) A U B [tex]\subseteq[/tex] B
2) B [tex]\subseteq[/tex] A U B

For case (1), I let x [tex]\in[/tex] A U B. Thus, either x [tex]\in[/tex] A or x [tex]\in[/tex] B. If x [tex]\in[/tex] A, then x [tex]\in[/tex] B. This means that A U B [tex]\subseteq[/tex] B.

For case (2), I let x [tex]\in[/tex] B. This is where I got stuck...
But if x ε B then isn't x ε A U B?
 
  • #3
dextercioby
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Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.
What you wrote holds only if A=B. Another scenario would be [itex] A\subseteq B \Rightarrow A\cup B = B [/itex]

which is trivial.
 
  • #4
vela
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Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.
From the other stuff you wrote, I'm assuming this is a typo and you meant if A ⊂ B, then A ∪ B = B.

In your attachment, you interpreted B ⊂ A ∪ B to mean B is a subset of A and a subset of B. This isn't correct. Say C ⊂ A ∪ B. If x∈C, that only means x∈A or x∈B. It's possible for some elements of C to be only in A and others to be only in B so that C is a subset of neither A nor B. So B ⊂ A ∪ B doesn't tell you anything about whether B is a subset of A or a subset of B. It just says all the elements in B are also elements in the union of A and B.
 

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