So A & D both are correct but A is more precise.

Consider two well-formed formulas in propositional logic

$F_1: P \Rightarrow \neg P$ $F_2: (P \Rightarrow \neg P) \lor ( \neg P \Rightarrow P)$

Which one of the following statements is correct?

- $F_1$ is satisfiable, $F_2$ is valid
- $F_1$ unsatisfiable, $F_2$ is satisfiable
- $F_1$ is unsatisfiable, $F_2$ is valid
- $F_1$ and $F_2$ are both satisfiable

### 6 Comments

$F_1: P \Rightarrow \neg P\equiv\neg P\vee \neg P\equiv\neg P$

When we put $P\equiv T$ it will $F$

and When we put $P\equiv F$ it will $T$

`It is called contingency.`

Always false called contradiction or unsatisfiable

Always true called valid or tautology

`At least`

one true called satisfiable.

$F_2: (P \Rightarrow \neg P) \lor ( \neg P \Rightarrow P)\equiv\neg P\vee P\equiv T$(Always true)

## 3 Answers

### 10 Comments

a) Satisfiable

If there is an assignment of truth values which makes that expression true.

b) UnSatisfiable

If there is no such assignment which makes the expression true

c) Valid

If the expression is Tautology

Here, P => Q is nothing but –P v Q

F1: P => -P = -P v –P = -P

F1 will be true if P is false and F1 will be false when P is true so F1 is Satisfiable

F2: (P => -P) v (-P => P) which is equals to (-P v-P) v (-(-P) v P) = (-P) v (P) =

Tautology

So, F1 is Satisfiable and F2 is valid

Option (a) is correct.

2."A formula is called valid if it is true in all the cases."

3.Valid formula or tautology are the same things.

4.Satisfiable and Contradiction are two opposite argument.

5.If formula is Satisfiable can not be Contradiction and vice-versa

**"A valid ***(true for every set of values) *** formula is always satisfiable **(*true for at least one value) **,* but a satisfiable formula may or may not be valid".

as F1 is not valid but satisfiable,

and F2 is valid(which implicitly means is satisfiable too).

therefore OPTION A is the complete solution, but OPTION D is not.