# Is the formula A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) proper?

I've read the formula created on the title on a maths' publication and also it does not appear proper for me :

**For the first component of the formula [A ∨ (B ∧ C) ] I have the adhering to feasible values : **

*A ; B and also C ; An and also B and also C (the Or is not unique) *

**For the 2nd component of the formula [ (A ∨ B) ∧ (A ∨ C) ] I have the adhering to values : **

*An and also A (A) ; An and also C ; B and also A ; B and also C ; ... ; An and also B and also An and also C (An and also B and also C) *

So I can have for the 2nd component of the formula *An and also B ; An and also C * which I can not get with the first component of the formula.

If I'm incorrect can someone please inform me just how and also offer some instances.

many thanks,

Bruno

Draw a Venn layout and also consider $A,B,C$ as collections. (containing True, False). After that you'll see that the formula is proper.

Or you can attract a fact table. http://en.wikipedia.org/wiki/Truth_table

Suppose you have $A \vee (B \wedge C)$. After that you either have $A$ or $B \wedge C$. If you have $A$, after that you have $A \vee B$ and also $A \vee C$. If, on the various other hand, you have $B \wedge C$, after that you have both $B$ and also $C$, in which instance you have both $A \vee B$ and also $A \vee C$. So $A \vee (B \wedge C) \Rightarrow (A \vee B) \wedge (A \vee C)$.

Entering the various other instructions, intend you have $(A \vee B) \wedge (A \vee C)$. So you have both $A \vee B$ and also $A \vee C$. Either you have $A$, or you do not. If you have $A$, after that you have $A \vee (B \wedge C)$. If, on the various other hand, you do not have $A$, after that the reality that you *do * have both $A \vee B$ and also $A \vee C$ indicates that you have to have both $B$ and also $C$. Consequently you have $B \wedge C$, therefore you have $A \vee (B \wedge C)$, i.e. $(A \vee B) \wedge (A \vee C) \Rightarrow A \vee (B \wedge C)$.

So we've developed that $A \vee (B \wedge C) \Rightarrow (A \vee B) \wedge (A \vee C)$ and also $(A \vee B) \wedge (A \vee C) \Rightarrow A \vee (B \wedge C)$. Consequently $A \vee (B \wedge C) = (A \vee B) \wedge (A \vee C)$.