Conjunctive Normal Form evaluates true when atleast half of the clauses are true. This is an Exam question.
Which of the Following is TRUE about formulae in Conjunctive Normal form?
-For any formula, there is a truth assignment for which at least half the clauses evaluate true.
-For any formula, there is a truth assignment for a which all the clauses evaluate to true.
-There is a formula such that for each truth assignment at most one fourth of the clauses evaluate to true.
-None
The answer given is :
"For any formula, there is a truth assignment for which at least half the clauses evaluate true."
To which I doubt too much!
This problem is asked earlier on this site itself but the accepted answer is somehow not at all satisfying to me.(may be I am a dumba**)
So here is what I interpreted step by step:

*

*Interpretation of CNF form:
E=(a+b+c)(a+b'+c)(a+b'+c')

(E is the expression/formula)
In short product of sum's of literals.


*Interpretation of term 'clauses':
Here in the above case there are 3 clauses. And In the total expression I am having 3 literals a, b, c.


*Interpretation of the problem :
Now First choice says:
"For any formula, there is a truth assignment for which at least half the clauses evaluate true."
Means (what i understood):
We are simply asked to find for E = 1 how many clauses should evaluate to true?
(As soon as I saw this question.. my first answer that popped In my head is: 100%..to which I agree still now)
Why / How 100%  ?
My answer would be : "how on Earth would some one multiply 0's and get 1?!".
What I feel those all clauses must evalute to true!!
Analogy:
π = X * Y *Z
(π is simply notation where the product is stored..!)
This is a product of 3 bits.
With 3 bits I can have 8 possibles combinations for these..
000,001,010,011,100,101,110,111
Now only 111 is the case when π would be 1(which is very obvious..even for a school kid )
Why? Bcoz x=y=z=1 so π=1 * 1 * 1 = 1
And in all 7 other options cases the product would be obviously zero.
That means for product of 3 term to be 'not zero' all three have to be 1 only!!
(Yes ofcourse right..or am i wrong here too!!?)
This is the basis for my ans being 100%
So, I think all the clauses must evalute to true, unless that happens, E can't be 1, it would be 0 only..(?)
Plz explain the option, and also comment on my interpretation ..whether that is correct in any sort?
Any help would be much regarded!
 A: As regards your attempt to interpret the problem, for your example
$$E=(a+b+c)(a+b'+c)(a+b'+c')$$
there are $3$ clauses, namely
$$a+b+c,\;\;a+b'+c,\;\;a+b'+c'$$
But the claim is not about assignments that make $E$ true.

Rather, the claim is that there is some assignment of truth values to the variables that makes at least half of the $3$ clauses true.

For this particular example, using $a=1$ and any truth values for $b,c$ makes all $3$ clauses true.

If instead we take the example
$$E=(a)(a')(a')$$
the truth value $a=0$ makes $2$ of the $3$ clauses true, but no truth value for $a$ makes all $3$ clauses true.

Let's consider the actual claim . . .

Claim:

If $P$ is a $\text{CNF}$ expression of the form $P=P_1\land \cdots \land P_k$ with variables in $\{X_1,...,X_n\}$, there is some truth assignment to the variables $X_1,...,X_n$ such that at least half of $P_1,...,P_k$ evaluate to true.

Proof:

Choose any truth assignment for the variables.

If at least half of $P_1,...,P_k$ evaluate to true, we're done.

If not, more than half of $P_1,...,P_k$ must evaluate to false, hence by negating the chosen truth value assignments for the variables, those of $P_1,...,P_k$ which evaluated to false will evaluate to true, so again we're done.
