COMPUTATIONAL-LOGIC Homework 3 Solved

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Let S be a binary predicate symbol, P and Q unary predicate symbols.

  • Find a natural deduction proof to show

xy(S(x,y) ∨ S(y,x)) ` ∃xyS(x,y).

  • Find a natural deduction proof to show

xyz(S(x,y)∧S(y,z) =⇒ S(x,z)),x¬S(x,x) ` ∀xy(S(x,y) =⇒ ¬S(y,x)).

  • Find a natural deduction proof to show

xy(S(x,y) ∨ S(y,x)),¬∃xS(x,x) ` ∃xy¬(x = y).

  • Show that there is no natural deduction proof for ∀x(P(x) ∨ Q(x)) ` ∀xP(x) ∨ ∀xQ(x).
  • Semantically show

x¬φ |= ¬∃xφ.

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