Set theory proof

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.

Last edited by a moderator:

PeroK
Homework Helper
Gold Member
2020 Award

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
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
Homework Helper
Gold Member
2020 Award
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
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
Staff Emeritus
Homework Helper
Gold Member

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
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!