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Unformatted text preview: valid. If they are valid, demonstrate their validity. You may use natural deduction or transformational proof. In both types of proofs, you may use the axioms and derived laws of set theory. If you use natural deduction, do not use any logical laws from transformational proof. In transformational proof, you may use the logical laws of predicate and propositional logic. If the argument is invalid, provide a counterexample and demonstrate that the argument is invalid. (a) R (  C  ) ∪ R (  C ′  ) = B where R : A ↔ B,C ⊆ A (b) ran(((( R ⊲ D ) ⊲ C ); id Y ) ∼ ) = dom( R ⊲ ( C ∪ D )) where R : X ↔ Y,C ⊆ Y,D ⊆ Y (c) ( B ⊳ R ) ∪ ( A ⊳ R ) = ( B − A ) ⊳ R where R : X ↔ Y,A ⊆ X,B ⊆ X (d) ( ∃ y • ( x,y ) ∈ ( R ∪ S )) ⊢ ( ∃ y • ( x,y ) ∈ ( R ⊕ S )) where R,S : X ↔ Y ( x is a constant.)...
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 Spring '08
 A
 Logic, transformational proof

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