(1) p∨∼(∼p→q)
=p∨∼(p∨q) Implication
=p∨(∼p∧∼q) De- Morgan's law
=(p∨∼p)∧(p∨∼q) Distributive law
=1∧(p∨∼q) Known tautology
=(p∨∼q) Dominance
=(∼q∨p) Commutative
=q→p Implication
(2)[(p→q)∧∼q]→∼p
=∼[(p→q)∧∼q]∨∼p Implication
=∼[(∼p∨q)∧∼q]∨∼p Implication
=∼[(∼p∧∼q)∨(q∧∼q)]∨∼p Distributive
= ∼[(∼p∧∼q)∨0]∨∼p Known contradiction
=∼[(∼p∧∼q)]∨∼p Dominance
=(p∨q)∨∼p De Morgan's Law
=(p∨∼p)∨q Associativity
=1∨q Known tautology
=1 Dominance
(3)[(p∨q)∧(p→∼r)∧r]→q
=[(p∨q)∧(∼p∨∼r)∧r]→q Implication
=[(p∨q)∧(∼p∧r)∨(∼r∧r)]→q Distributive
=[(p∨q)∧(∼p∧r)∨0]→q Known contradiction
=[(p∨q]∧(∼p∧r)]→q Dominance
=∼[(p∨q)∧(∼p∧r)]→q Implication
=∼(p∨q)∨∼(∼p∧r)∨q De Morgan
=∼(p∨q)∨(p∨∼r)∨q De Morgan
=∼(p∨q)∨(p∨q)∨∼r Associativity
=1∨∼r Known tautology
=1 Dominance
(4)(p∨∼q)∧(p∨q)
=p∨(∼q∧q) Distributive law
=p∨0 Known contradiction
=p Dominance
(5)∼[p→∼(p∧q)]
=∼[∼p∨∼(p∧q)] Implication
=p∧(p∧q) De Morgan's Law
=(p∧p)∧q Associative
=p∧q Idempotent
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