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 (p → r) Λ (q → r)] → r

Order Options

[(p V q) Λ (p → r) Λ (q → r)] → 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 (p → r) Λ (q → r)] → 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 (p → r) Λ (q → r)] → 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