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Which of the following propositions is tautology?

  1. $(p\lor q)\to q$
  2. $p\lor (q\to p)$
  3. $p\lor (p\to q)$
  4. Both (B) and (C)
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Option $(C)$

$p\vee (p\rightarrow q) \equiv p\vee \neg p \vee q \equiv T \vee q \equiv T$
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3 Answers

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ANS :C 

 

p∨(p→q)

p q P->Q p∨(P->Q)

T

T T T
T F F T
F T T T
F F T T

 

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To prove that the given wff isn't a tautology we need to get False value. For A -> B, we need to get T -> F case.

A.  q=F, p v q need to be T, q is F, p we can set as T. T v F = T , so T -> F. Hence not a tautology.

B. p = F, q=T. p v (q -> p) => F v (T -> F) => F

C. p = F, p -> q need to be false. But as p is T, we can never get it at F. Suppose we set p = T, then expression is always T.

So C is correct.
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A tautology is a proposition that is always true for every value of its propositional values.

Note: we can solve such types of questions by either using the truth table method or applying logical equivalences rules. Here I use the truth table method.

Option 1) : not tautology.

$p$ $q$ $p\lor q$

$(p\lor q)\to q$

T T T T
T F T F
F T T T
F F F T

Option 2): Not tautology.

$p$ $q$ $q\to p$

$p \lor (q \to p)$

T T T T
T F T T
F T F F
F F T T

option 3): it is a tautology.

$p$ $q$ $p\to q$ $p \lor(p\to q)$
T T T T
T F F T
F T T T
F F T T

Option $(C)$ is correct.

Answer:

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