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FORMAL LANGUAGE & AUTOMATA THEORYState and prove the pumping lemma for regular languages.20217mFormal Language & Automata TheoryState and prove pumping lemma for regular sets.20217mDiscrete 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 < v20228mDiscrete Mathematics(a) Prove the following: (i) There is a real number x such that x > 1 and 2^x > x^{10}. (ii) There is an integer n such that 2n^2 - 5n + 2 is prime.202210m
PreviousProve the following by using the principle of mathematical induction for all n \in N: 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2NextCheck the validity of the following argument all integers are rational numbers. Some integers are powers of 5. There fore, some rational numbers are powers of 5
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