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# 数学代写|实分析代写Real Analysis代考|Cartesian product of sets

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## 数学代写|实分析代写Real Analysis代考|Cartesian product of sets

Let $A$ and $B$ be non-empty sets. The Cartesian product of $A$ and $I$ denoted by $A \times B$, is the set defined by
$$A \times B={(a, b): a \in A, b \in B}$$
$A \times B$ is the set of all ordered pairs $(a, b)$, the first element of the pa being an element of $A$ and the second being an element of $b$.

Let $A_1, A_2, \ldots, A_n$ be a finite collection of non-empty sets. The Cart sian product of the collection, denoted by $A_1 \times A_2 \times \cdots \times A_n$, is the si defined by
$$A_1 \times A_2 \times \cdots \times A_n=\left{\left(a_1, a_2, \ldots, a_n\right): a_i \in A_i, i=1,2, \ldots, n\right} .$$
In particular if $A_1=A_2=\cdots=A_n=A$, then the Cartesian proc uct of the collection, denoted by $A^n$, is the set of all ordered $n$ tuple $\left{\left(a_1, a_2, \ldots, a_n\right): a_i \in A, i=1,2, \ldots, n\right}$

Let $A$ and $B$ be two non-empty sets. Intuitively, a relation $\rho$ betwee $A$ and $B$ is a rule that associates some or all the elements of $A$ with sor element or elements of $B$.

Definition. Let $A$ and $B$ be two non-empty sets. A relation $\rho$ betwee $A$ and $B$ is a subset of $A \times B$. If the ordered pair $(a, b) \in \rho$ then th element $a$ of the set $A$ is said to be related to the element $b$ in $B$ by tr relation $\rho$. If $(a, b) \in(A \times B)-\rho$, then $a$ is said to be not related to $b \mathrm{t}$ the relation $\rho$.

## 数学代写|实分析代写Real Analysis代考|Order relation on a: set

Definition. Partially ordered set.
A relation $\rho$ on a non-empty set $X$ is said to be a partial order relation if $\rho$ is reflexive, anti-symmetric and transitive.

A set $X$ equipped with a partial order relation $\rho$ is said to be a partially ordered set (or a poset) and it is denoted by $(X, \rho)$.

Note. A partial order relation is commonly denoted by the symbol $\leq$ or $\geq$ and read in usual manner. Thus $a \leq b$ is read as ” $a$ is less than or equal to $b$ “. A partially ordered set $X$ with a partial order $\leq$ is denoted by $(X, \leq)$.
Examples (continued).

A relation $\rho$ on a set $X$ is said to be a strict order relation if it is anti-symmetric and transitive and for which $(a, a) \notin \rho$ for all $a \in X$.
If a partial order be denoted by $\leq$, then the corresponding strict order is denoted by $<$.

Two elements $a$ and $b$ in a partially ordered set are said to be comparable if one of them is related to the other, i.e., one of the relations $a \leq b$, $b \leq a$ must hold. In a partially ordered set there may exist elements $a$ and $b$ which are not comparable. For example, in Ex.3, the integers 4 and 6 are not comparable, because neither is a divisor of the other.

If a partial order relation satisfies a fourth condition that ‘any two elements are comparable’ then it is called a total order relation.

## 数学代写|实分析代写Real Analysis代考|Cartesian product of sets

$$A \times B={(a, b): a \in A, b \in B}$$
$A \times B$是所有有序对$(a, b)$的集合，其中pa的第一个元素是$A$的元素，第二个元素是$b$的元素。

$$A_1 \times A_2 \times \cdots \times A_n=\left{\left(a_1, a_2, \ldots, a_n\right): a_i \in A_i, i=1,2, \ldots, n\right} .$$

## MATLAB代写

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