Exam_2_solutions

# Exam_2_solutions - EXAM 2 Unless noted otherwise let L be...

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EXAM 2 Unless noted otherwise let L be the PC 0 language (with equality) generated by the following: H (unary relation symbol), L (binary relation symbol), and a,b,c (constant symbols). 1. True or False (2pts each) The following refer to the semantics for L . (1) ¬ ( a b ) ∧ ¬ ( b c ) | = ¬ ( a c ) False . Let I be an interpretation such that U = { 0 , 1 } , ¯ a = 0, ¯ b = 1, and ¯ c = 0. I | = ¬ ( a b ) ∧ ¬ ( b c ) but I | = a c . (2) {{{}}} and {{}} have the same number of elements. True . {{}} is the only element of {{{}}} and {} is the only element of {{}} . (3) If φ ∈ L and a a | = φ , then φ is a logical truth. True . Let I be an interpretation. From the semantics for ‘ ’ it is clear that I | = a a . Hence, since a a | = φ , it follows that I | = φ . (4) If I = h U, ¯ H, ¯ L, ¯ a, ¯ b, ¯ c i is an interpretation of L such that I | = ¬ ( a b ), then U has at least two elements. True

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Exam_2_solutions - EXAM 2 Unless noted otherwise let L be...

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