Definition [first-order formulas] Let S be a signature. The (first-order) formulas are strings of symbols over the alphabet of S, inductively defined as follows: (1) If t0, t1 are terms, then t0 = t1 is an formula. (2) If t0, . . . ,tn−1 are terms, and R is an n-ary relation symbol in S, then R(t0, . . . ,tn−1) is an formula (3) If ϕ is an formula, then ¬ϕ is an formula (4) If ϕ0, ϕ1 are formulas, then (ϕ0 ∨ ϕ1), (ϕ0 ∧ ϕ1), (ϕ0 → ϕ1), (ϕ0 ↔ ϕ1) are formulas (5) If ϕ is an formula and x is a variable, then ∀x : ϕ and ∃x : ϕ are both formulas。翻译理解分析这些内容

这段文字是关于一阶逻辑中公式的定义。在一个给定的符号集合S（称为signature）中，一个公式是由S中的符号构成的字符串。这些公式可以通过以下方式归纳定义：

(1) 如果t0，t1是项，则t0 = t1是一个公式。
(2) 如果t0，…，tn−1是项，且R是S中的一个n元关系符号，则R(t0,…,tn−1)是一个公式。
(3) 如果ϕ是一个公式，则¬ϕ也是一个公式。
(4) 如果ϕ0，ϕ1是公式，则(ϕ0 ∨ ϕ1)、(ϕ0 ∧ ϕ1)、(ϕ0 → ϕ1)和(ϕ0 ↔ ϕ1)都是公式。
(5) 如果ϕ是一个公式，x是一个变量，则∀x : ϕ和∃x : ϕ都是公式。

其中，项是由符号集合S中的变量、常量和函数符号构成的一个表达式。关系符号是S中的一种符号，表示一种n元关系。公式是一种用于描述S中对象之间关系的表达式，可以理解为一种命题。其中，∀x表示“对于任意x”，∃x表示“存在一个x”，→表示“蕴含”，∧表示“合取”，∨表示“析取”，↔表示“等价”。这些逻辑符号可以用于表达不同的逻辑关系和命题。

相关问题

Write a program that calculates the circumference and area ofa Sem radius.

Sure, here's a Python program that calculates the circumference and area of a circle with a given radius:

import math

radius = float(input("Enter the radius of the circle: "))

circumference = 2 * math.pi * radius
area = math.pi * radius ** 2

print("The circumference is:", circumference)
print("The area is:", area)

In this program, we first prompt the user to enter the radius of the circle. We then use the math module to calculate the circumference and area of the circle, using the formulas:

- Circumference = 2 * pi * radius
- Area = pi * radius^2

Finally, we print out the results to the user.

3. (简答题)For converting temperature from Celsius to Fahrenheit, or from Fahrenheit to Celsius, we can use the following formulas. -------------------------------------------------- Celsius To Fahrenheit: Fahrenheit = ( Celsius * 9 / 5 ) + 32; Fahrenheit To Celsius: Celsius = ( Fahrenheit - 32 ) * 5 / 9; -------------------------------------------------- Write a program to convert temperature from Fahrenheit to Celsius and vice versa using switch case. In the program, there are three options given to the user. The first choice is to convert the temperature from Celsius to Fahrenheit. The second choice is to convert the temperature from Fahrenheit to Celsius. And the last one is to exit without testing any one of the queries. The program should work as follows: ---------------------------------------------------------- Select Code： For Fahrenheit To Celsius For Celsius To Fahrenheit Enter Your Choice: 1 Enter The Value of Fahrenheit Temperature: 90 Celsius Temperature: 32.2 ---------------------------------------------------------- Select Code： 1. For Fahrenheit To Celsius 2. For Celsius To Fahrenheit Enter Your Choice:2 Enter The Value of Celsius Temperature: 30 Fahrenheit Temperature: 86.0 ---------------------------------------------------------- Select Code： 1. For Fahrenheit To Celsius 2. For Celsius To Fahrenheit Enter Your Choice:3 Wrong Choice.....Try Again!!! ----------------------------------------------------------

以下是使用 switch case 实现温度转换的程序：

#include <iostream>
using namespace std;

int main() {
    int choice;
    float temp, result;
    
    while (true) {
        cout << "Select Code:\n";
        cout << "1. For Fahrenheit To Celsius\n";
        cout << "2. For Celsius To Fahrenheit\n";
        cout << "3. Exit\n";
        cout << "Enter Your Choice: ";
        cin >> choice;

        if (choice == 3) {
            cout << "Exiting...";
            break;
        }

        switch (choice) {
            case 1:
                cout << "Enter The Value of Fahrenheit Temperature: ";
                cin >> temp;
                result = (temp - 32) * 5 / 9;
                cout << "Celsius Temperature: " << result << "\n";
                break;
            case 2:
                cout << "Enter The Value of Celsius Temperature: ";
                cin >> temp;
                result = (temp * 9 / 5) + 32;
                cout << "Fahrenheit Temperature: " << result << "\n";
                break;
            default:
                cout << "Wrong Choice.....Try Again!!!\n";
                break;
        }
    }

    return 0;
}

运行程序后，用户可以选择输入选项，然后输入相应的温度值，程序会输出转换结果。如果用户选择退出选项，则程序结束。