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Discrete 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.202210mDiscrete Mathematics(a) Let D = \{-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36\} Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false. (i) \forall x \in D, if x is odd, then x > 0 (ii) \forall x \in D, if x is less than 0, then x is even (iii) \forall x \in D, if x is even, then x \le 020226mDiscrete 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 MathematicsUse Cantor's diagonal argument to prove that set F of all functions f: (0,1) \rightarrow R has larger Cardinality than |R|.20237m
PreviousState and prove Division algorithm theorem well-ordering principle.NextA grocery store employee is stocking apples. Each apple is a different color. There are 10 apples left in the box and the employee pulls out 2 of them at random. What is the probability that the employee pulls out one pink apple and yellow apple?
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