Propositional Logic

Question

#Discrete_Maths
#Logic

I have a doubt in the following concept.
Suppose we're given:
(A v B) ^ ( B v C)...when we open the brackets, do we apply distributive law or just open it..as in
1. A v B ^ B v C = A v B ^ C
2. ((A v B) ^ B) v ((A v B) ^ C) ( by distributive law)

Which one of the above is correct and why? Also, when exactly do we apply distributive law? I see that sometimes, the brackets are opened just like that and in other cases, distributive law is used. How to differentiate what to do when?

Please help!

Answer 1

(A v B) ^ ( B v C)
Here precedence of bracket is more.
Hence A v B & B v C will be evaluated 1st. Then  ^

As per your 1st statement
 A v B ^ B v C  ,   here precedence of   ^ is more.
It means   A v (B ^ B) v C    which is A v B  v C.

(A v B) ^ ( B v C)   &  A v B  v C  are not same.
Draw truth table to be clear.
Draw truth table ((A v B) ^ B) v ((A v B) ^ C) too..  You will get same as (A v B) ^ ( B v C)

Comments

But then while proving the an argument ( inference), one of the ways is that we take conjunction of all the statements and apply intution or maybe inference rules right. Let me give an example.
Suppose there are two premises--
A-->B
B-->C
We have to prove A-->C
Now, if we apply distributive rule:
(A-->B) ^ (B-->C) ===
((A-->B) ^ ~B ) v ((A-->B) ^ C)
I don't think we'll get the correct answer.
On the other hand, if we just open the brackets, we'll get the answer, as in, A-->B ^ B-->C==
 ~A v B ^ ~B v C ==
~A v C v F ==
~A v C==
A--> C

How's this possible. This is what is confusing me.

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Whatever distributive property gives that is correct.
You can not open bracket. Precedence of bracket is more than AND. So first bracket should be evaluated then anything else.

& Are you sure ?
A-->B
B-->C
(A-->B) ^(B-->C)   equivalent to A-> C   ?? No.
( (A-->B) ^(B-->C)  ) -->  A-> C