1) Simplify the left side of the equivalence
((P∧Q∧R)→S∧(R→(P∨Q∨S)))=(((P∧Q∧R)∨S)∧(R∨(P∨Q∨S)))=(P∨Q∨R∨S)∧(R∨P∨Q∨S)=(R∨S)∨((P∨Q)∧(P∨Q))=(R∨S)∨(P∧P∨Q∧P∨P∧Q∨Q∧Q)=(R∨S)∨(0∨Q∧P∨P∧Q∨0)=(R∨S)∨(Q∧P∨P∧Q)=R∨S∨(P⊕Q)
Simplify the right side of the equivalence:
R∧(P↔Q)→S=R∧(P→Q)∧(Q→P)→S=R∧(P∨Q)∧(Q∨P)→S=R∧(P∨Q)∧(Q∨P)∨S=R∨P∧Q∨Q∧P∨S=R∨(P⊕Q)∨S We have:
R∨S∨(P⊕Q)≡R∨(P⊕Q)∨S
Equivalence is performed.
Answer: Equivalence is performed
2) Simplify the left side of the equivalence
((P∧Q)→R)∧(Q→(S∨R))=((P∧Q)∨R)∧(Q∨(S∨R))=(P∨Q∨R)∧(Q∨S∨R)=Q∨R∨P∧S
Simplify the right side of the equivalence
Q∧(S→P)→R=Q∧(S→P)∨R=Q∧(S∨P)∨R=Q∨(S∨P)∨R=Q∨S∧P∨R
We have
Q∨R∨P∧S≡Q∨S∧P∨R
Equivalence is performed.
Answer: Equivalence is performed