Question

Consider the following arguments. If an argument is valid, then present a proof sequence; otherwise, prove that the argument is invalid. You are forbidden to use truth tables to justify your answers (but, you may use them otherwise).

((p → r) ∨ (q → r)) → ((p ∨ q) → r)

((q → r) ∧ (p → (q ∨ r))) → (p → r)

((p → (q ∧ r)) ∧ (s → r) ∧ (s → t)) → (t → p)

Answer 1

4) a) Given ((p   r) V (q   r))   ((p V q)   r)

((¬p V r) V (¬q V r))   (¬(p V q) V r)  { Law of Implies (PQ) = ¬PVQ }

¬((¬p V r) V (¬q V r)) V (¬(p V q) V r)  { Law of Implies (PQ) = ¬PVQ }

(¬(¬p V r) ¬(¬q V r)) V (¬(p V q) V r) {By De Morgan's law ¬ (P V Q) = ¬P ¬Q }

((¬(¬p) ¬r) (¬(¬q) ¬r) V ((¬p ¬q) V r) {By De Morgan's law ¬ (P V Q) = ¬P ¬Q }

(p   ¬r) (q   ¬r) V ( r V (¬p ¬q)) {By De Morgan's law ¬ (¬ P) = P }

(p   q) (¬r   ¬r) V ( r V (¬p ¬q)) {Commutative law (P Q) = (Q P)}

(p   q) (¬r) V ( r V (¬p ¬q))  { we know that ¬P ¬P = ¬P }

(p   q) (¬r V r) V (¬p ¬q) {Associative law (P V Q) V R = PV (Q V R)}

(p   q) (T) V (¬p ¬q) { we know that ¬P V P = T }

((q   p) (¬p ¬q) V (T) {Commutative law (P Q) = (Q P)}

(q   p) (¬p ¬q) V (T)

q   (p ¬p) ¬q V (T) {Associative law (P Q) R = P (Q R)}

q   (F) ¬q V (T) { we know that P ¬P = F }

q   ¬q (F) V (T)   {Commutative law (P Q) = (Q P)}

(q   ¬q) (F) V (T)   {Associative law (P Q) R = P (Q R)}

(F) (F) V (T)   { we know that P ¬P = F }

(F F) V (T)   { we know that F F = F }

(F) V (T)   { we know that F V T = T }

  (T)

The given argument is Valid

4) b) Given ((q   r)   (p   (q V r)))   (p   r)

((¬q V r)   (¬p V (q V r)))   (¬p V r) { Law of Implies (PQ) = ¬PVQ }

¬((¬q V r)   (¬p V (q V r))) V (¬p V r) { Law of Implies (PQ) = ¬PVQ }

(¬(¬q V r) V ¬(¬p V (q V r))) V (¬p V r)   {By De Morgan's law ¬ (P Q) = ¬P V ¬Q }

((¬(¬q) ¬r)) V (¬(¬p) ¬(q V r))) V (¬p V r) {By De Morgan's law ¬ (P V Q) = ¬P ¬Q }

((q ¬r)) V (p   (¬q ¬r))) V (¬p V r) {By De Morgan's law ¬ (¬ P) = P }

q ¬r V (p   ¬q) ¬r V (r V ¬p) {Associative law (P Q) R = P (Q R)}

q (p   ¬q) V ¬r ¬r V (r V ¬p)  {Commutative law (P V Q) = (Q V P)}

p (q   ¬q) V (¬r ¬r) V (r V ¬p)  {Commutative law (P Q) = (Q P)}

p (F) V (¬r) V (r V ¬p) { we know that P ¬P = F }

p (F) V (¬r V r) V ¬p {Associative law (P V Q) V R = PV (Q V R)}

p (F) V (T) V ¬p  { we know that ¬P V P = T }

  (F) V (T)   p V ¬p {Commutative law (P Q) = (Q P)}

  (F V T) (p V ¬p) {Associative law (P V Q) V R = PV (Q V R)}

  (T) (T)  { we know that ¬P V P = T }

  (T)

The given argument is Valid

4) c) Given ((p    (q r))   (s    r)   (s   t))   (t   p)

((p' V (q r))   (s' V r)   (s' V t))   (t' V p)  { Law of Implies (PQ) = ¬PVQ }

(((p' V (q r))   (s' V r)   (s' V t)))' V (t' V p)  { Law of Implies (PQ) = ¬PVQ }

((p' V (q r))' V (s' V r)' V (s' V t)' ) V (t' V p)

(((p')' (q r)') V ((s')' r') V ((s')' t') ) V (t' V p)

((p (q' V r')) V (s r') V (s t') ) V (t' V p)

((p (q' V r')) V (r' s) V (s t') ) V (t' V p)

p' q' V (r' V r') (s V s) (t' V t') V p {Associative law (P V Q) V R = PV (Q V R)}

p' q' V (r') (s) (t') V p   { we know that ¬P V P = T }

p' p V q' V (r') (s) (t')   {Commutative law (P V Q) = (Q V P)}

( p' p) V q' V (r') (s) (t') {Associative law (P Q) R = P (Q R)}

(F) V q' V (r') (s) (t')  { we know that P P' = F }

q' V (r') V (F) (s) (t') {Commutative law (P V Q) = (Q V P)}

q' V (r') V (F   s) (t')  {Associative law (P Q) R = P (Q R)}

q' V (r') V (F) (t') { we know that P F = F }

q' V (r') V (F t')

q' V (r') V (F) { we know that P' F = F }

q' V (r' V F)

q' V (r')

The given argument is Invalid