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# Rules of inference

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Discrete mathematics, 6 rules of inference

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### Rules of inference

1. 1. RULES OF INFERENCE (CONCLUSION)1. Rule of conjunctive simplificationThis rule states that, P is true whenever PΛQ is true.Symbolically it is PΛQ ∴PExplanation: pΛq is similar to AND logic of Digital electronics. Its truth value is 1 only ifp and q both are 1. So we can say p is true whenever PΛQ is true.Proof by truth table:P Q PΛQ0 0 00 1 01 0 01 1 12. Rule of disjunctive amplificationThis rule states that P⋁Q is true whenever P is true.Symbolically it is P ∴ P⋁QExplanation: P⋁Q is similar to OR logic of Digital electronics. Its truth value is 1 if any ofthe inputs ( p or q )is 1. So we can say P⋁Q is true whenever P is true, irrespective ofthe value of Q.Proof By Truthtable:P Q PVQ0 0 00 1 11 0 11 1 1
2. 2. 3. Rule of Hypothetical syllogism:This rule states that P⟶R is true whenever P⟶ Q is true and Q⟶R is true.Symbolically it is P⟶Q Q⟶R ∴ P⟶RExplanation: P⟶Q is true, it states that P is not 1 and Q is not 0 concurrently.Same way, Q⟶R is true, it states that Q is not 1 and R is not 0 at the same time.(Because A⟶B is a conditional statement whose value is 0 only in one case which iswhen A is1 and B is 0)So we can say both these statements can take any othercombination other than 1 and 0.Let’s say P is 1 ie Q is also 1.Going to statement (b), if Q is 1 from above statement, R must be 1.And if P is 1 and Q is 1 and R is also 1. This implies P⟶R is also one that is only ourconclusion.Proof By TruthtableP Q R P⟶Q Q⟶R P⟶R0 0 0 1 1 10 0 1 1 1 10 1 0 1 0 00 1 1 1 1 11 0 0 0 1 11 0 1 0 1 11 1 0 1 0 01 1 1 1 1 1
3. 3. 4. Rule of Disjunctive syllogismThis rule states that Q is true whenever P⋁Q is true and P is true.Symbolically it is P⋁Q (a) P (b) ∴QExplanation: P⋁Q behaves the same way as OR gate in digital electronics ie, if any of thetwo inputs is 1 the output is 1. So if we know that PVQ is 1(from (a)) and P is 1 thatmeans P is 0. So to support (a), Q has to be one only then the output of PVQ will be 1.This implies Q is 1 and that is our conclusion.Proof By TruthtableP Q P⋁Q P0 0 0 10 1 1 11 0 1 01 1 1 05. Rule of Modus Pones( Rule of detachment)This rule states that Q is true whenever P is true and P⟶Q is true.Symbolically it is P (a) P⟶Q (b) ∴QExplanation: P⟶Q is a conditional statement in discrete mathematics, which is falseonly in one case that is when p is true and q is false (or P is 1 and Q is 0 then p⟶q is 0and in all other cases it is true.)Here we know that p is true, from (a)And P⟶Q is true (from (b)) ie Q must be true that is the conclution.
4. 4. Proof By TruthtableP Q P⟶Q0 0 10 1 11 0 01 1 16. Rule of Modus TollensThis rule states that ~P is true whenever P⟶Q is true and ~Q is true.Symbolically it is P⟶Q ……………… (a) ~Q ……………….. (b) ∴~PExplanation: P⟶Q is a conditional statement in discrete mathematics, which is falseonly in one case that is when p is true and q is false (or P is 1 and Q is 0 then p⟶q is is0 and in all other cases it is true)We know that ~Q is 1from (b) ie Q is 0 ( we know that ~Q is inverse of Q)The above condition P⟶Q can be true only if P is 0 because if P is 1 and Q is 0 theoutput will be 0.So it signifies that p is 0 and ~P is 1 that is our conclusion.Proof By TruthtableP Q ~Q ~P P⟶Q0 0 1 1 10 1 0 1 11 0 1 0 01 1 0 0 1