1. Prove using set algebra

Prove using set algebra:

A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B)

2. Re: Prove using set algebra

Originally Posted by mathman11
Prove using set algebra:

A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B)

https://proofwiki.org/wiki/Set_Diffe...form_Partition

3. Re: Prove using set algebra

Hello, mathman11!

Prove using set algebra: . $A \cup B \;=\; (A \cap B') \cup (A' \cap B) \cup (A \cap B)$

There are two rules for which I do not know the names.

I will call them: . $\begin{Bmatrix}S \cup S' \:=\:U && \text{Property 1} \\ S \cap U \:=\:S && \text{Property 2} \end{Bmatrix}$

Consider the right side.

$\begin{array}{cccccc}1. & (A\cap B') \cup (A' \cap B) \cup (A \cap B) && 1. & \text{Given} \\ \\ 2. & (A\cap B') \cup \left[(A' \cap B) \cup (A \cap B)\right] && 2.& \text{Associative} \\ \\ 3. & (A \cap B') \cup\left[(A' \cup A) \cap B\right] && 3. & \text{Distributive} \\ \\ 4. & (A \cap B') \cup (U \cap B) && 4. & \text{Prop. 1} \\ \\ 5. & (A \cap B') \cup B && 5. & \text{Prop. 2} \\ \\ 6. & (A \cup B) \cap (B' \cup B) && 6. & \text{Distributive} \\ \\ 7. & (A \cup B) \cap U && 7. & \text{Prop. 1} \\ \\ 8. & A\cup B && 8. & \text{Prop. 2} \end{array}$

4. Re: Prove using set algebra

thanks Soroban this was really helpful.

Do you have any tips as to how to approach questions like the one you just did?