MathsGee Answers is Zero-Rated (You do not need data to access) on: Telkom | Dimension Data | Rain | MWEB
First time here? Checkout the FAQs!
x
MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB

0 like 0 dislike
15 views
What is the multiplication principle in set theory?
in Mathematics by Diamond (74,866 points) | 15 views

1 Answer

0 like 0 dislike
Best answer
If you have two finite sets $A$ and $B$, where $A$ has $M$ elements and $B$ has $N$ elements, then $A \times B$ has $M \times N$ elements. This rule is called the multiplication principle and is very useful in counting the numbers of elements in sets. The number of elements in a set is denoted by $|A|$, so here we write $|A|=M,|B|=N$, and $|A \times B|=M N$. In the above example, $|A|=3,|B|=2$, thus $|A \times B|=3 \times 2=6$. We can similarly define the Cartesian product of $n$ sets $A_{1}, A_{2}, \cdots, A_{n}$ as
$$
A_{1} \times A_{2} \times A_{3} \times \cdots \times A_{n}=\left\{\left(x_{1}, x_{2}, \cdots, x_{n}\right) \mid x_{1} \in A_{1} \text { and } x_{2} \in A_{2} \text { and } \cdots x_{n} \in A_{n}\right\}
$$
The multiplication principle states that for finite sets $A_{1}, A_{2}, \cdots, A_{n}$, if
$$
\left|A_{1}\right|=M_{1},\left|A_{2}\right|=M_{2}, \cdots,\left|A_{n}\right|=M_{n}
$$
then
$$
\left|A_{1} \times A_{2} \times A_{3} \times \cdots \times A_{n}\right|=M_{1} \times M_{2} \times M_{3} \times \cdots \times M_{n}
$$
An important example of sets obtained using a Cartesian product is $\mathbb{R}^{n}$, where $n$ is a natural number. For $n=2$, we have
$$
\begin{aligned}
\mathbb{R}^{2} &=\mathbb{R} \times \mathbb{R} \\
&=\{(x, y) \mid x \in \mathbb{R}, y \in \mathbb{R}\}
\end{aligned}
$$
$\mathbb{R}^{3}=\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ and so on.
by Diamond (74,866 points)

Related questions

0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked May 26, 2020 in Data Science & Statistics by MathsGee Diamond (74,866 points) | 56 views
1 like 0 dislike
1 answer
asked May 27 in Mathematics by Teddy Bronze Status (8,688 points) | 16 views
0 like 0 dislike
1 answer
asked May 27 in Mathematics by Teddy Bronze Status (8,688 points) | 11 views
0 like 0 dislike
1 answer
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true

MathsGee provides answers to subject-specific educational questions for improved outcomes.



On MathsGee Answers, you can:


1. Ask questions
2. Answer questions
3. Comment on Answers
4. Vote on Questions and Answers
5. Donate to your favourite users

MathsGee Tools

Math Worksheet Generator

Math Algebra Solver

Trigonometry Simulations

Vectors Simulations

Matrix Arithmetic Simulations

Matrix Transformations Simulations

Quadratic Equations Simulations

Probability & Statistics Simulations

PHET Simulations

Visual Statistics

ZeroEd Search Engine

Other Tools

MathsGee ZOOM | eBook