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Question: prove the given expression is a tautology by developing a...

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Prove the given expression is a tautology by developing a series of logical equivalence to demonstrate that it is logically equivalent to T.

[(p V q) Λ (pr) Λ (qr)] → r

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_________________ [(p V q) Λ (pr) Λ (qr)] → r = [(p V q) Λ (p V q) → r] by logical equivalence

_________________ ( (p V q) Λ r ) → r by identity law

_________________    ((p V q) Λ r ) → r = ¬((p V q) Λ r ) V r by logical equivalence

_________________ ( ¬p Λ ¬q) V T by negation law

_________________ [ T  V ((p V q) Λ r )] → r by negation law

_________________   [(p V q) Λ (pr) Λ (qr)] → r   by associative law

_________________ [((p V q) Λ (p V q) V (¬(p V q) Λ r )] → r   by distributive law

_________________ [((p V q) Λ (¬(p V q) V (p V q) Λ r ) → r   by distributive law

_________________ T by domination law

_________________ [(p V q) Λ (pr) Λ (qr)] → r

_________________ ( ¬p Λ  ¬q) V ( ¬r V r) by associative law

_________________ [(p V q) Λ ((p V q) → r ) → r = [(p V q) Λ (¬(p V q)V r)] → r by logical equivalence

[F V ((p V q) Λr)] → r by negation law

   (( ¬p Λ  ¬q) V ¬r ) V r   by De Morgan's law

( ¬p Λ  ¬q) V F by negation law   

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