GO Classes 2023 | Weekly Quiz 3 | Question: 15

Answer 1

Answer: B, C, D

Given $F$ and $G$ are propositional formula.

• A. It is false because for $F \lor G$ to be a tautology there is a possiblity that when $F$ is false $G$ is true and vice versa, in this case both $F$ and $G$ are contingency yet $F \lor G$ is tautology.

 

• A. If $F$ is tautology and $F \rightarrow G$ is tautology then $G$ must be a tautology because if $G$ is either contingency and takes False or it is contradiction then $F \rightarrow G$ will not be a tautology.

 

100. Let’s simplify this 
     $(F \rightarrow G) \lor (F \rightarrow \lnot G) = \lnot F \lor G \lor \lnot F \lor \lnot G$ as we can see $G \lor \lnot G$ will always gives True and $T \lor anything$ is true, hence this is a tautology.

 

500. For this let’s consider $F$ is either contingency or a tautology in this case $F$ can be True, now if $F$ is true then for $(F \rightarrow G) \land (F \rightarrow \lnot G)$ to be a tautology both $G$ and $\lnot G$ must be true that is not possible obviously, so for $(F \rightarrow G) \land (F \rightarrow \lnot G)$ to be tautology $F$ must be a contradiction.

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Find the Detailed Video Solution: Propositional Formula Question - Complete Analysis​​​​​​​