Write a formal proof that proves $ (A \cap B) \cup(A\cap B') = A $
Our professor showed us the following proof:
- $\forall x \in (A \cap B) \cup(A\cap B')$
- $\text{----}$Suppose $ x\in B$
- $\text{----}$Then $ x\in A \cap B$
- $\text{----}$So $ x\in A $
- $\text{----}$Suppose $ x\in B'$
- $\text{----}$Then $ x\in A \cap B'$
- $\text{----}$So $ x\in A $
- Since $ x\in B$ or $ x\in B'$, $x \in A$
I tried understanding this proof several times and can't understand the jump between step 2 and 3 and between 5 and 6. Here are my explanations for the other steps:
- $\forall x \in (A \cap B) \cup(A\cap B')$ (Assumption)
- $\text{----}$Suppose $ x\in B$ (Assumption)
- $\text{----}$Then $ x\in A \cap B$ (????)
- $\text{----}$So $ x\in A $ (Simplification)
- $\text{----}$Suppose $ x\in B'$ (Assumption)
- $\text{----}$Then $ x\in A \cap B'$ (????)
- $\text{----}$So $ x\in A $ (Simplification)
- Since $ x\in B$ or $ x\in B'$, $x \in A$ (Resolution)
I just do not understand how you can conclude that $x$ is in $A\cap B$ if it's in $B$