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

###### Question details

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