- #1

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.

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