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Unformatted text preview: Nanyang Technological University Mid-Term Test 2009 MAS 111 Foundations of Mathematics Hints Instructions to the candidates • The examination consists of FIVE (5) problems. • Answer all questions. The marks for each question are indicated at the end of each question. • All calculations and answers should be accompanied by appropriate explanations. Problem 1 1. Show by constructing truth-tables that u ∧ ( v ∨ w ) and ( u ∧ v ) ∨ w are not equivalent and that ( u ∧ ( v ∨ w )) → (( u ∧ v ) ∨ w ) is a tautology. (10 marks) 2. Prove by applying logical equivalences that r → ( p → q ) is logically equivalent to ( p ∧ r ) → q . (So do not use truth-tables.) (10 marks) Hints : Easy. Problem 2 1. Let U = Z be a universal set, P ( x ) be the statement “ x > 1 ”, and Q ( x ) be the statement “ x < 6 ”. (8 marks) Determine which of the following are true and which are false. (1) ∀ x ( P ( x ) ∨ Q ( x )) , (2) ( ∀ x P ( x )) ∨ ( ∀ x Q ( x )) ....
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This note was uploaded on 01/23/2011 for the course SPMS MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.
- Spring '10