cs245-asn5 - valid If they are valid demonstrate their...

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CS 245 Winter 2009 Assignment 5 Due: Thu 12 Mar 2009 10am in the CS245 Drop Boxes 35 marks You must submit your assignment with a cover page produced using makeCover , otherwise you will lose 2 marks. Assignments are to be completed individually. The policy on academic offenses is on the course web page. 1. (9 marks) Formalize the following sentences in set theory. Do not use types, quantifiers, or set comprehension. Use only the following sets and relations in your formulas: Dwellings – the set of dwelling People – the set of people Houses – the set of houses Students – the set of students owns : People Dwellings The relationship between people and the dwellings that they own. rents : People Dwellings The relationship between people and the dwellings that they rent. where Students People , Houses Dwellings . Sentences to formalize: (a) All houses that are rented are owned. (b) Not every student owns a house. (c) Students who rent houses do not own any dwelling. 1
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2. (26 marks) Determine whether the following arguments in set theory are valid or in-
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Unformatted text preview: valid. If they are valid, demonstrate their validity. You may use natural deduction or trans-formational 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 transfor-mational 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 argu-ment 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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