Set theory proof

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Homework Statement



Prove that ##A \cup (A \cap B) = A##

Homework Equations



In the previous exercise, we proved:

Let A, B be sets. Then, the following statements are equivalent:

1) ##A \subseteq B##
2) ##A \cup B = B##
3) ##A \cap B = A##

The Attempt at a Solution



The proof of ##A \cup (A \cap B) = A## according to the teacher was: we can use this previous exercise to show that ##A \cap B = A## Then the problem becomes that ##A \cup (A \cap B) = A \cap A = A##

However, we are not given that ##A \subseteq B##. Hence, we cannot apply this previous exercise. So, is the teacher's proof wrong?

Btw, can someone verify my proof:

Proof:

To show that ##A \cup (A \cap B) = A##, we show that ##A \cup (A \cap B) \subseteq A \land A \subseteq A \cup (A \cap B)##

1) Take ##x \in A \Rightarrow x\in A \cup (A \cap B)## by definition of union.
We deduce that ##A \subseteq A \cup (A \cap B)##

2) Take ##x \in A \cup (A \cap B) \Rightarrow x \in A \lor x \in A \cap B##
##\Rightarrow x \in A \lor (x \in A \land x \in B)##
##\Rightarrow x \in A## (logic argument: ##p \lor (p \land q) \Rightarrow p##)
We deduce that ##A \cup (A \cap B) \subseteq A##

QED.
 
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  • #2
PeroK
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Homework Statement



Prove that ##A \cup (A \cap B) = A##

Homework Equations



In the previous exercise, we proved:

Let A, B be sets. Then, the following statements are equivalent:

1) ##A \subseteq B##
2) ##A \cup B = B##
3) ##A \cap B = A##

The Attempt at a Solution



The proof of ##A \cup (A \cap B) = A## according to the teacher was: we can use this previous exercise to show that ##A \cap B = A## Then the problem becomes that ##A \cup (A \cap B) = A \cap A = A##


However, we are not given that ##A \subseteq B##. Hence, we cannot apply this previous exercise. So, is the teacher's proof wrong?

What do you think?
 
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  • #3
member 587159
What do you think?

I think he's wrong. Can you verify the proof that I made please? (I edited my post)
 
  • #4
PeroK
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I think he's wrong. Can you verify the proof that I made please? (I edited my post)

I think your teacher might have meant ##A \cap B \subseteq A## hence ##A \cup (A \cap B) = A##

Your proof looks a bit over-elaborate to me.
 
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  • #5
member 587159
I think your teacher might have meant ##A \cap B \subseteq A## hence ##A \cup (A \cap B) = A##

Your proof looks a bit over-elaborate to me.

I'm not sure what the teacher meant. Next time I'll see him I'll ask what he meant. Thanks for your help.
 
  • #6
SammyS
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Homework Statement



Prove that ##A \cup (A \cap B) = A##

Homework Equations



In the previous exercise, we proved:

Let A, B be sets. Then, the following statements are equivalent:

1) ##A \subseteq B##
2) ##A \cup B = B##
3) ##A \cap B = A##

The Attempt at a Solution



The proof of ##A \cup (A \cap B) = A## according to the teacher was: we can use this previous exercise to show that ##A \cap B = A## Then the problem becomes that ##A \cup (A \cap B) = A \cap A = A##

However, we are not given that ##A \subseteq B##. Hence, we cannot apply this previous exercise. So, is the teacher's proof wrong?
You do know that ##A \cap B\subseteq A\,,\ ## Right?

The proof pretty much follows from there.
 
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  • #7
member 587159
You do know that ##A \cap B\subseteq A\,,\ ## Right?

The proof pretty much follows from there.

Wow didn't see that. Guess that happens when it's late. Thanks!
 

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