Explanation Given: A and B are regular sets. By Kleene’s theorem,a set is regular if and only if it is recognized by a finite-state automaton. Since, A is regular, so there isa finite-state automaton M that accepts A. By just changing the final states to the non-final states and the non-final states to final states of this machine we obtain a machine that accepts A ¯ . Thus, by Kleen’s theorem, A ¯ is regular. Similarly B ¯ is also regular. Now, by the definition of regular sets, if two sets are regular then their union is also regular. Thus, A ¯ ∪ B ¯ is also regular

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Author: ROSEN

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 13, Problem 24SE

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If A and Bare regular sets then, A∩B is also regular.

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Chapter 13, Problem 23SE

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An improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = xn +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution . İs Zn+1 = Yn + h · k₁. Then we find the slope at the predicted new point k₁ = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = = Yn + 1½ ½ (k1 + k₂). Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx = x - 0.5y, y(0.5) = 9. We let xo = 0.5 and yo 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k₂ 0 0.59-48 -3.25 ♡ <+ help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = = help (formulas) Thus the actual value of the…

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Chapter 13 Solutions

DISCRETE MATHEMATICS+ITS APPL. (LL)-W/A

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To show that: If A and Bare regular sets then, A ∩ B is also regular.

Solution Summary: The author explains Kleene's theorem that a set is regular if it is recognized by finite-state automatons. If two sets are regular, their union is also regular.

Concept explainers

To show that:

If A and Bare regular sets then, A∩B is also regular.

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Chapter 13 Solutions

To show that: If A and Bare regular sets then, A ∩ B is also regular.

Solution Summary: The author explains Kleene's theorem that a set is regular if it is recognized by finite-state automatons. If two sets are regular, their union is also regular.

Concept explainers

To show that: If A and Bare regular sets then, A∩B is also regular.

Want to see the full answer?

Chapter 13 Solutions

To show that:

If A and Bare regular sets then, A∩B is also regular.