tut3 - Problems to Week 4 Tutorial MACM 101(Fall 2011 1...

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Problems to Week 4 Tutorial — MACM 101 (Fall 2011) 1. Consider the universe of all polygons with three or four sides, and define the following predicates for this universe: A ( x ): all interior angles of x are equal; E ( x ): x is a equilateral triangle; H ( x ): all sides of x are equal; P ( x ): x has an interior angle that exceeds 180 ; Q ( x ): x is a quadrilateral; R ( x ): x is a rectangle; S ( x ): x is a square; T ( x ): x is a triangle. Translate each of the following statements into an English sentence, and determine whether the statement is true or false. (a) x ( Q ( x ) T ( x )); (b) x ( T ( x ) P ( x )); (c) x ( Q ( x ) ∧ ¬ R ( x )); (d) x ( H ( x ) E ( x )); (e) x ( S ( x ) ( A ( x ) H ( x ))). 2. Let P ( x, y ) , Q ( x, y ) denote the following predicates P ( x, y ) : x 2 y Q ( x, y ) : x + 2 < y. If the universe for each of x, y consists of all real numbers, determine the truth value for each of the following statements (a) P ( - 3 , 8) Q (1 , 3); (b) P ( 1 2 , 1 3 ) ∨ ¬ Q ( - 2 , - 3); (c) P (1 , 2) ↔ ¬ Q (1 , 2). 3. Let P ( x ) , Q ( x ), and R ( x ) denote the following predicates P ( x ) : x 2 - 8 x + 15 = 0 Q ( x ) : x is odd R ( x ) : x > 0 .
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