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Chapter 12: Problem 3

Let \(\Sigma=\left\\{1,2,^{*}, /\right\\}\) and let \(L\) be the set of all strings over \(\Sigma\) obtained by writing first a number (1 or 2), then a second number (1 or 2), which can be the same as the first one, and finally an operation ( \({ }^{*}\) or / where * indicates multiplication and / indicates division). Then \(L\) is a set of postfix, or reverse Polish, expressions. List all the elements of \(L\) between braces, and evaluate the resulting expressions.

Short Answer

Expert verified
The set L of postfix expressions and their evaluations is as follows: L = { 11*: 1, 11/: 1, 12*: 2, 12/: 0.5, 21*: 2, 21/: 2, 22*: 4, 22/: 1 }

Step by step solution

01

Identify all possible elements of set L

To find all possible elements of set L, we need to combine all the given numbers and operations in the prescribed order. This will result in 8 different expressions.
02

List all elements of set L

By combining all possible combinations of numbers and operations, we will obtain the following strings within the set L: 1. 11* 2. 11/ 3. 12* 4. 12/ 5. 21* 6. 21/ 7. 22* 8. 22/
03

Evaluate the expressions

Now, we will evaluate the expressions one by one: 1. 1*1 = 1 2. \( \frac{1}{1} \) = 1 3. 1*2 = 2 4. \( \frac{1}{2} \) = 0.5 5. 2*1 = 2 6. \( \frac{2}{1} \) = 2 7. 2*2 = 4 8. \( \frac{2}{2} \) = 1 Lastly, we can rewrite the set L by including the results of each expression: L = { 11*: 1, 11/: 1, 12*: 2, 12/: 0.5, 21*: 2, 21/: 2, 22*: 4, 22/: 1 }

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluation of Postfix Expressions
Postfix expressions, also known as Reverse Polish notation (RPN), are an expression format where operators follow their operands. For example, the postfix expression for "1 + 2" would be "12+". This format eliminates the need for parentheses that are traditionally used to specify the order of operations in infix notation.
When evaluating postfix expressions, the process is straightforward:
  • Read the expression from left to right.
  • Push operands (numbers) onto a stack as you encounter them.
  • When an operator is encountered, pop the necessary number of operands from the stack, apply the operator, and push the result back onto the stack.
This systematic approach ensures that operations are executed in the correct order and is well-suited for evaluation by computers.
Reverse Polish Notation
Reverse Polish notation (RPN) is an efficient way to represent mathematical expressions without the need for parentheses or rules of precedence. This lack of ambiguity in RPN makes it especially useful in programming languages and computational mathematics.
In RPN, each operator follows its operands, allowing immediate execution of operations as they are encountered. To illustrate, the infix expression "((1 + 2) * 3)" is expressed in postfix notation as "12+3*".
  • The simplicity of RPN reduces the computational steps needed to evaluate expressions.
  • It aligns naturally with stack-based calculations, which helps to maintain the correct order of operations.
  • RPN can simplify the process of programming mathematical operations because it removes the usual operator precedence issues.
Overall, RPN provides a clear, concise method of expression manipulation.
Discrete Mathematics
Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. It includes a wide range of topics such as logic, set theory, graph theory, and combinatorics.
In the context of evaluating postfix expressions, discrete mathematics principles, such as logic and set theory, come into play:
  • Logic helps in understanding the processes of reasoning in evaluating expressions.
  • Set theory, like the formation of a set of expressions, provides a foundational framework for handling collections of distinct objects.
  • Understanding these concepts aids in constructing algorithms that efficiently process and evaluate expressions.
These concepts provide students with the tools necessary to explore more in-depth topics in mathematics and computer science, such as algorithm development and cryptography.

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Most popular questions from this chapter

Prove that if two states of a finite-state automaton are \(k\)-equivalent for some integer \(k\), then those states are \(m\)-equivalent for all nonnegative integers \(m

A simplified telephone switching system allows the following strings as legal telephone numbers: a. A string of seven digits that does not start with \(00,01,10\) or 11 ( \(a\) local call string), b. A 1 followed by a three-digit area code string (any digit except 0 or I followed by a 0 or 1 followed by any digit) followed by a seven-digit local call string. c. A 0 alone or followed by a three-digit area code string plus a seven-digit local call string, Design a finite-state automaton to recognize all the legal telephone numbers in (a), (b) and (c). Include an "error state" for invalid telephone numbers.

a. Let \(A\) be a finite-state automaton with input alphabet \(\Sigma\), and suppose \(L(A)\) is the language accepted by \(A\). The complement of \(L(A)\) is the set of all strings over \(\Sigma\) that are not in \(L(A)\). Show that the complement of a regular language is regular by proving the following: If \(L(A)\) is the language accepted by a finite-state automaton \(A\), then there is a finite-state automaton \(A^{\prime}\) that accepts the complement of \(L(A)\). b. Show that the intersection of any two regular languages is regular as follows: First prove that if \(L\left(A_{1}\right)\) and \(L\left(A_{2}\right)\) are languages accepted by automata \(A_{1}\) and \(A_{2}\), respectively, then there is an automaton \(A\) that accepts \(\left(L\left(A_{1}\right)\right)^{c} \cup\left(L\left(A_{2}\right)\right)^{c}\). Then use one of De Morgan's laws for sets, the double complement law for sets, and the result of part (a) to prove that there is an automaton that accepts \(L\left(A_{1}\right) \cap L\left(A_{2}\right)\).

\((x y)\left(\left(\left(x^{*}\right) y\right)^{*}\right) \mid\left(((y x) \mid y)\left(y^{*}\right)\right)\)

All United States social security numbers (which consist of three digits, a hyphen, two digits, another hyphen, and finally four more digits), where the final four digits start with a 3 and end with a 6 .
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