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Discrete Mathematics(b) Indicate which of the following statements are true and which are false. Justify your answers as best you can: (i) \forall x \in \mathbf{Z}^+, \exists y \in \mathbf{Z}^+ such that x = y+1 (ii) \forall x \in \mathbf{Z}, \exists y \in \mathbf{Z} such that x = y+1 (iii) \exists x \in \mathbf{R} such that \forall y \in \mathbf{R}, x = y+1 (iv) \forall x \in \mathbf{R}^+, \exists y \in \mathbf{R}^+ such that xy = 1 (v) \forall x \in \mathbf{R}, \exists y \in \mathbf{R} such that xy = 1 (vi) \forall x \in \mathbf{Z}^+ \text{ and } \forall y \in \mathbf{Z}^+, \exists z \in \mathbf{Z}^+ \text{ such that } z = x-y (vii) \forall x \in \mathbf{Z} \text{ and } \forall y \in \mathbf{Z}, \exists z \in \mathbf{Z} \text{ such that } z = x-y (viii) \exists u \in \mathbf{R}^+ \text{ such that } \forall v \in \mathbf{R}^+, uv < v20228mFormal Language & Automata TheoryWhich of the following statements is/are False? A. For every non-deterministic Turing machine, there exists an equivalent deterministic Turing machine. B. Turing recognizable languages are closed under union and complementation. C. Turing decidable languages are closed under intersection and complementation. D. Turing recognizable languages are closed under union and intersection. (i) A and D only (ii) A and C only (iii) B only (iv) C only20212mFORMAL LANGUAGE & AUTOMATA THEORYWhich of the following statements is/are false? A. For every nondeterministic TM, an equivalent deterministic TM exists. B. Turing recognizable languages are closed under union and complementation. C. Turing decidable languages are closed under intersection and complementation. D. Turing recognizable languages are closed under union and intersection. (i) A and D only (ii) A and C only (iii) B only (iv) C only20212mDiscrete MathematicsWhich of the following statements regarding sets is false? (i) A \cap A = A (ii) A \cup A = A (iii) A - (B \cap C) = (A - B) \cup (A - C) (iv) (A \cup B)' = A' \cup B'20222m
PreviousThe number of edges in a regular graph of degree 46 and 8 vertices is (i) 347 (ii) 230 (iii) 184 (iv) 186Next(b) Indicate which of the following statements are true and which are false. Justify your answers as best you can: (i) \forall x \in \mathbf{Z}^+, \exists y \in \mathbf{Z}^+ such that x = y+1 (ii) \forall x \in \mathbf{Z}, \exists y \in \mathbf{Z} such that x = y+1 (iii) \exists x \in \mathbf{R} such that \forall y \in \mathbf{R}, x = y+1 (iv) \forall x \in \mathbf{R}^+, \exists y \in \mathbf{R}^+ such that xy = 1 (v) \forall x \in \mathbf{R}, \exists y \in \mathbf{R} such that xy = 1 (vi) \forall x \in \mathbf{Z}^+ \text{ and } \forall y \in \mathbf{Z}^+, \exists z \in \mathbf{Z}^+ \text{ such that } z = x-y (vii) \forall x \in \mathbf{Z} \text{ and } \forall y \in \mathbf{Z}, \exists z \in \mathbf{Z} \text{ such that } z = x-y (viii) \exists u \in \mathbf{R}^+ \text{ such that } \forall v \in \mathbf{R}^+, uv < v
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