practice_final_2010

# practice_final_2010 - PRACTICE FINAL 1 True or False(2 pts...

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Unformatted text preview: PRACTICE FINAL 1. True or False? (2 pts each) L is a first-order language (with equality). M and M are interpretations of L . X M is the domain of M . X M is the domain of M . ψ ( x ) is a formula in L that has x as its only free variable. θ [ x : t ] is obtained from θ by replacing every free occurrence of x by t . • If X M has more elements than X M and X M is finite, then there is a φ ∈ L such that M | = φ and M | = ¬ φ . • If A and B are L-definable subsets of X M , then A × B is an L-definable subset of X M × X M . • If M | = ∀ xψ ( x ), then X M is the subset of X M that is defined by ψ ( x ). • If M | = ∃ xα ∧ ∃ xβ , then M | = ∃ x ( α ∧ β ). • If a and b are constant symbols in L and M | = θ [ x : a ] ↔ θ [ x : b ] for all θ , where θ has x as its only free variable, then M | = a ≈ b . 2. Explain (5 pts each) Explain two of your answers from Section 1. 3. Inductive Definition (5 pts each) Using the successor function, give an inductive definition of the set { ( q,r,s ) | q,r,s ∈ N and q + r ≤ s } . Show that your set contains (2 , 1 , 3). 4. Evaluate Sentences (2 pts each) Let L be contain three binary relation symbols: R 1 , R 2 , and R 3 . Let M be the following interpretation of L : • X M = { (...
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practice_final_2010 - PRACTICE FINAL 1 True or False(2 pts...

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