GO Classes 2023 | Weekly Quiz 3 | Question: 10

Answer 1

We'll first discuss what the option says -
Option A : p is valid (all rows true) and satisfiable (at least one row true). This is only possible if p is valid ie all rows true (valid is a subset of satisfiable).
Option B : p is invalid (at least one row false) and unsatisfiable (all rows false). This is only possible if p is unsatisfiable ie all rows false (unsatisfiable is a subset of invalid).
Option C : p is invalid (at least one row false) and satisfiable (at least one row true)
Option D : p is valid (all rows true) and unsatisfiable (all rows false). How can any proposition be true for all rows and false for all rows at the same time. This option doesn't make sense.

Now, coming back to the problem -  we've been told that p is the shortest formula (atomic preposition) we don't know whether it's true or false. But we know it can be any one of them.

Option A says p is valid, but we know that is not the case. Option B says p is unsatisfiable, but again we know that is not the case.

What we know is p can be true (which makes p satisfiable) and p can be false (which makes p invalid).

Answer : C

Answer 2

The formula $p$ evaluates to $F$ under the evaluation $p \rightarrow F$, and therefore the formula is invalid. It evaluates to $T$ under the valuation $p \rightarrow T$ and therefore the formula is satisifiable.

Answer 3

Before answering this question let us try to understand the following terms.

1. Valid:- Same as a tautology. That means always true
2. Satisfiable:- That means it can be made True. So every valid is satisfiable. 
3. invalid:- Not a tautology. 
4. Unsatisfiable:- Always false. 

 Now coming back to the question, “ p” is said to be an atomic proposition. This means p can either be true or false.

Thus valid and unsatisfiable ruled out. So option 3 is the answer.