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FORMAL LANGUAGE & AUTOMATA THEORYShow that the function: f(x,y) = x + y is primitive recursive.20217mDesign & Analysis of AlgorithmsProve that if f_1(n) = O(g_1(n)) and f_2(n) = O(g_2(n)), then f_1(n) + f_2(n) = O(g_1(n) + g_2(n)).201914mDesign and Analysis of Algorithms Solve the following recurrence by successive substitution method: f(1) = 1 \quad \text{if } n=1 f(n) = 3f(n/2) + 6 \quad \text{if } n > 120227mDiscrete 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 < v20228m
PreviousShow given grammar over alphabet \{a, b\}, verify whether it is ambiguous or unambiguous: S \to aSa \mid bSb \mid a \mid b \mid \epsilonNextConstruct a PDA that accepts the language L = \{a^{2n}bc \mid n \ge 0\} by final state and empty stack.
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