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Which of the following well-formed formulas are equivalent?

  1. $P \rightarrow Q$
  2. $\neg Q \rightarrow \neg P$
  3. $\neg P \vee Q$
  4. $\neg Q \rightarrow P$
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7 Answers

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This method is useful only when we unable to apply the equivalence formulas. It’s like a simplified version of the truth table method.

Two or more compound propositions are equivalent if they have the same truth value for every possible combination of the truth value of its atomic propositions(recall the last two columns of being the same in the truth table method)

Here we want to do that in reverse.First assume all the given compound propositions to be True or all the given compound to be false

Then find out the possible combinations of truth values of atomic propositions present in the compound propositions

In this Question I take all given propositions to be False then find all the possible combinations of  P and Q (in order)

solution

Since first 3 propositions have same set of combinations of P and Q we can conclude that they are equivalent(no need to find possible set of P, Q values for getting True here)

We can verify the same by choosing all propositions to be true as  shown below

solution 2

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Here we will first try to use the properties of implication and check which one’s are similar,

(a) : P → Q

(b) : P → Q (by property of contrapositive)

© : P → Q (we know, P → Q = P’ V Q)

(d) : Q V P

 

Therefore, a,b,c are equivalent.
Answer:

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