Proof of A Union of A Intersection B Equals A

[member 587159]

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.

 

[PeroK]

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?

 

[member 587159]

What do you think?

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

 

[PeroK]

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.

 

[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.

 

[SammyS]

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.

 

[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!