GO Classes Weekly Quiz 5 | Propositional Logic | Question: 16

Answer 1

Given: $F1$, $F2$, and $F3$ are propositional formulae, over some set of propositional variables, such that $F1 \vee F2 \rightarrow F3$ is a contradiction.

Propositional formula $F1 \vee F2 \rightarrow F3$ can be a contradiction (Always $False$) when $F3$ is a Contradiction and $F1 \vee F2$ is Tautology.

• A. Both F1, F2 are tautologies. False because over some set of variables, the propositional formula can be $F1=True$, $F2=False$ and $F1=False$, $F2=True$ is both satisfying the condition which is $F1 \vee F2$ is a tautology, $F1$, $F2$ need not be a tautology.
• B. At least one of $F1$, $F2$ is a tautology. False, same reason as option A.
• C. $F3$ is a contradiction. True.
• D. $F1 \vee F2$ is a tautology. True.

Therefore, conditions that are $necessarily\: true$ such that $F1 \vee F2 \rightarrow F3$ is a contradiction are options C and D.

Comments

How can you say, “False same as option A”. If both are false, then the prop formula is tautology so atleast one of F1 and F2 must be tautology, to be a contradiction.

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By False I meant the option is false and no, at least one of F1 or F2 need not be a tautology. Consider these 2 instances (F1=True, F2=False) and (F1=False, F2=True), if both the cases are possible for some combination over some set of variables, then F1 and F2 are not tautology still the disjunction F1 $\vee$ F2 is True making the formula F1 $\vee$ F2 $\rightarrow$ F3 a contradiction.

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A good answer @sankalps.

Detailed Video Solution: Detailed Video Solution, Weekly Quiz 5 Question 16 & Question 7

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Thank you sir!

Answer 2

F1, F2 and F3 are Propositional Expressions and not propositional variables.

(F1 or F2) → F3 is contradiction(always false). So, (F1 or F2) is tautology(always true) and F3 is contradiction(always false).

...(propositional variables)...	F1	F2	F3	F1 or F2	F1 or F2 → F3
….	T	T	F	T	F
….	T	F	F	T	F
….	F	T	F	T	F

From this table we can see that, it is not necessary that F1 is Tautology. Similarly, it is not necessary that F2 is Tautology. But (F1 or F2) is tautology.

Answer 3

As per my understanding option B is a bi-implication of option D. i.e

F1+F2 = 1 <---> F1=1 or F2=1

Here is how I thought,

For contradiction let us assume “atleast any one of F1 or F2 is tautology is not a necessary condition for F1+F2→ F3 being a contradiction” which means neither of F1,F2 being tautology doesn't affect F1+F2→ F3 being a contradiction.

So let us assume F1 and F2 are contingencies which means both F1 and F2 can take false value for some combination of their dependent propositional variables, so if F1=FALSE and F2=FALSE then F1+F2=FALSE which means FALSE->F3 is always TRUE

Therefore it shows neither of F1,F2 being tautology affects F1+F2->F3 being a contradiction hence our assumption “atleast any one of F1 or F2 is tautology is not a necessary condition for F1+F2→ F3 being a contradiction” is an invalid assumption.

@Deepak+poonia please correct me if I am going wrong somewhere

Comments

@Deepak Poonia

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@swapnilbanerjee 

Detailed Video Solution: Weekly Quiz 5 Detailed Video Solutions

Answer 4

This question is a lookalike of question 7 from the same quiz set but is a little different.

For F1 V F2 = F3 = False (viz contradiction)

it should follow T → F = F

so, F1 V F2 = T and F3 = F. 

Thus, option (d) is correct and option (c ) is correct.

Not sure about (b) yet as it seems correct to me but the answer key does not match. 

 

After watching Deepak sir’s explanation, I changed my answer so that anybody reading it does not get confused.

A propositional variable can be called tautology ONLY WHEN it gives True as the truth value ALWAYS.

In this question, F1 V F2 = T in three cases, TF, FT, TT. So we can’t say that F1 is tautology or F2 is tautology as its values are changing. However, we can say that the disjunction is always true so it will be a tautology.

Thus, at least one of F1 and F2 is not tautology.

Comments

The take away from this question should be the counter example for option B.

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Detailed Video Solution: Weekly Quiz 5 Detailed Video Solutions 

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@Deepak Poonia

This is really a good question. It will clarify the logic behind the difference between truth value and tautology and the proper definition of propositional formula/expression. 🙏