A variable is sometimes thought of as a mathematical “John Doe” because you can use it as a placeholder when you want to talk about something but either (1) you imagine that it has one or more values but you don’t know what they are, or (2) you want whatever you say about it to be equally true for all elements in a given set, and so you don’t want to be restricted to considering only a particular, concrete value for it. To illustrate the first use, consider asking Is there a number with the following property: doubling it and adding 3 gives the same result as squaring it? In this sentence you can introduce a variable to replace the potentially ambiguous word “it”: Is there a number $x$ with the property that $2 x+3=x^2$ ? The advantage of using a variable is that it allows you to give a temporary name to what you are seeking so that you can perform concrete computations with it to help discover its possible values. To emphasize the role of the variable as a placeholder, you might write the following: Is there a number $\square$ with the property that $2 \cdot \square+3=\square^2$ ? The emptiness of the box can help you imagine filling it in with a variety of different values, some of which might make the two sides equal and others of which might not.
数学代写|离散数学代写Discrete Mathematics代考|Writing Sentences Using Variables
Use variables to rewrite the following sentences more formally. a. Are there numbers with the property that the sum of their squares equals the square of their sum? b. Given any real number, its square is nonnegative. Solution a. Are there numbers $a$ and $b$ with the property that $a^2+b^2=(a+b)^2$ ? Or: Are there numbers $a$ and $b$ such that $a^2+b^2=(a+b)^2$ ? Or: Do there exist any numbers $a$ and $b$ such that $a^2+b^2=(a+b)^2$ ? b. Given any real number $r, r^2$ is nonnegative. Or: For any real number $r, r^2 \geq 0$. $O r:$ For every real number $r, r^2 \geq 0$.
现代博弈论始于约翰-冯-诺伊曼（John von Neumann）提出的两人零和博弈中的混合策略均衡的观点及其证明。冯-诺依曼的原始证明使用了关于连续映射到紧凑凸集的布劳威尔定点定理，这成为博弈论和数学经济学的标准方法。在他的论文之后，1944年，他与奥斯卡-莫根斯特恩（Oskar Morgenstern）共同撰写了《游戏和经济行为理论》一书，该书考虑了几个参与者的合作游戏。这本书的第二版提供了预期效用的公理理论，使数理统计学家和经济学家能够处理不确定性下的决策。
