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# 经济代写| SETS, FUNCTIONS, AND REAL NUMBERS 微观经济学代写

## 经济代写

2.1. Sets, tuples, and Cartesian products. The above definition of a decision situation in strategic form contains several important mathematical concepts. In line with the philosophy of this book to explain mathematical concepts wherever they arise for the first time, we offer some comments on sets, tuples, the Cartesian product of sets, functions, and real numbers.
First, a set is any collection of objects that can be distinguished from each other. A set can be empty in which case we use the symbol $\emptyset$. The objects are called elements. In the above definition, we have the sets $S, W$, $\mathbb{R}$, and $S \times W$ (the latter being a Cartesian product).

DEFINITION II.3 (SET AND SUBSET). Let $M$ be a nonempty set. $A$ set $N$ is called a subset of $M$ (denoted by $N \subseteq M$ ) if and only if every element from $N$ is contained in $M$. We use curly brackets {} to indicate sets. Two sets $M_{1}$ and $M_{2}$ are equal if and only if $M_{1}$ is a subset of $M_{2}$ and $M_{2}$ is a subset of $M_{1}$. We define strict inclusion $N \subset M$ by $N \subseteq M$ and $M \nsubseteq N$.
The reader will note the pedantic use of “if and only if” in the above definition. In definitions (!), it is quite sufficient to write “if” instead of “if and only if” (or the shorter “iff”).

Sets need to be distinguished from tuples where the order is important:
DEFINITION II.4 2. SETS, FUNCTIONS, AND REAL NUMBERS
13 DEFINITION II.4 (TUPLE). Let $M$ be a nonempty set. A tuple on $M$ is an ordered list of elements from $M$. Elements can appear several times. $A$ tuple consisting of $n$ entries is called an $n$-tuple. We use round brackets () to denote tuples. Two tuples $\left(a_{1}, \ldots, a_{n}\right)$ and $\left(b_{1}, \ldots, b_{m}\right)$ are equal if they have the same number of entries, i.e., if $n=m$ holds, and if the respective entries are the same, i.e., if $a_{i}=b_{i}$ for all $i=1, \ldots, n$.

Oftentimes, we consider tuples where each entry stems from a particular set. For example, $S \times W$ is the set of tuples $(s, w)$ where $s$ is a strategy from $S$ and $w$ a state of the world from $W$.

DEFINITION II.5 (CARTESIAN PRODUCT). Let $M_{1}$ and $M_{2}$ be nonempty sets. The Cartesian product of $M_{1}$ and $M_{2}$ is denoted by $M_{1} \times M_{2}$ and defined by
$$M_{1} \times M_{2}:=\left{\left(m_{1}, m_{2}\right): m_{1} \in M_{1}, m_{2} \in M_{2}\right} .$$
EXERCISE II.2. Let $M:={1,2,3}$ and $N:={2,3}$. Find $M \times N$ and depict this set in a two-dimensional figure where $M$ is associated with the abscissa (x-axis) and $N$ with the ordinate ( $y$-axis).
2.2. Injective and surjective functions. We now turn to the concept of a function. The payoff function $u: S \times W \rightarrow \mathbb{R}$ is our first example.
DEFINITION II. 6 (FUNCTION). Let $M$ and $N$ be nonempty sets. A function $f: M \rightarrow N$ associates with every $m \in M$ an element from $N$, denoted by $f(m)$ and called the value of $f$ at $m$. The set $M$ is called the domain (of $f)$, the set $N$ is the range (of $f)$ and $f(M):={f(m): m \in M}$ the image (of $f$ ). A function is called injective if $f(m)=f\left(m^{\prime}\right)$ implies $m=m^{\prime}$ for all $m, m^{\prime} \in M$. It is surjective if $f(M)=N$ holds. A function that is both injective and surjective is called bijective.

EXERCISE II.3. Let $M:={1,2,3}$ and $N:={a, b, c}$. Define $f: M \rightarrow N$ by $f(1)=a, f(2)=a$ and $f(3)=c$. Is $f$ surjective or injective?

When describing a function, we use two different sorts of arrows. First, we have $\rightarrow$ in $f: M \rightarrow N$ where the domain is left of the arrow and the range to the right. Second, on the level of individual elements of $M$ and $N$, we use $\mapsto$ to write $m \mapsto f(m)$. For example, a quadratic function may be

2.1。集合、元组和笛卡尔积。上述以战略形式定义的决策情境包含几个重要的数学概念。根据本书解释数学概念首次出现的哲学，我们对集合、元组、集合、函数和实数的笛卡尔积提出了一些评论。

13 定义 II.4（元组）。令$M$ 为非空集。 $M$ 上的元组是 $M$ 中元素的有序列表。元素可以出现多次。由 $n$ 个条目组成的 $A$ 元组称为 $n$-tuple。我们使用圆括号 () 来表示元组。两个元组 $\left(a_{1}, \ldots, a_{n}\right)$ 和 $\left(b_{1}, \ldots, b_{m}\right)$ 相等，如果它们有相同条目数，即，如果 $n=m$ 成立，并且如果各个条目相同，即对于所有 $i=1，\ldots，n$，如果 $a_{i}=b_{i}$。

$$M_{1} \times M_{2}:=\left{\left(m_{1}, m_{2}\right): m_{1} \in M_{1}, m_{2} \in M_ {2}\右} 。$$

2.2.内射函数和满射函数。我们现在转向函数的概念。支付函数 $u: S \times W \rightarrow \mathbb{R}$ 是我们的第一个例子。