Bowtie

The Mathematical Symbol "Bowtie (⋈)"

The ⋈ Symbol in Mathematics: Natural Join and Relations

The ⋈ symbol in mathematics, resembling a bowtie, is not as widely recognized as symbols like "+" or "-". However, in specific domains, especially within relational database theory and certain areas of set theory, it plays an essential role. This lesson covers two primary contexts where the ⋈ symbol finds its application, providing two illustrative examples for each.

1. Relational Database Theory

In relational databases, the ⋈ symbol represents the "natural join" of two tables. It's a binary operation that combines two tables based on common attributes.

• Example 1: If there are two tables, `Employees` and `Departments`, with a common attribute `DepartmentID`, the natural join of these tables can be expressed as: `Employees ⋈ Departments`. This operation results in a combined table including rows where the `DepartmentID` matches in both original tables.
• Example 2: For tables `Orders` and `Customers` having a shared attribute `CustomerID`, the expression `Orders ⋈ Customers` denotes the natural join, merging order details with respective customer information based on the `CustomerID`.

2. Set Theory and Relations

In certain areas of set theory and relations, the ⋈ symbol represents a specific kind of product or combination of relations.

• Example 1: Given two relations \( R \) and \( S \) on sets \( A \) and \( B \) respectively, the expression \( R ⋈ S \) can symbolize a specific type of product or composite relation between the two.
• Example 2: If \( R \) is a relation from set \( A \) to \( B \) and \( S \) is a relation from \( B \) to \( C \), then the composite relation \( R ⋈ S \) might denote pairs from \( A \) to \( C \) that can be formed by combining pairs in \( R \) and \( S \).

In summary, while the ⋈ symbol might not be an everyday mathematical notation for many, its significance in relational database theory and set theoretic relations is undeniable. Recognizing its meaning in these contexts can lead to deeper insights into the structures and operations discussed.

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Codes for the ⋈ Symbol

The Symbol	⋈	copy
Alt Code	Alt 8904	copy
HTML Code	⋈	copy
HTML Entity	⋈	copy
CSS Code	\22C8	copy
Hex Code	⋈	copy
Unicode	U+22C8	copy

How To Insert the ⋈ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ⋈ symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8904. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML Text	Output
<b>My symbol: &#8904;</b>	My symbol: ⋈

(Method 4) Use the HTML Entity Code (for webpages).

HTML Text	Output
<b>My symbol: &bowtie;</b>	My symbol: ⋈

(Method 5) Use the CSS Code (for webpages).

CSS and HTML Text	Output
<style> span:after { content: "\22C8";} </style> <span>My symbol:</span>	My symbol: ⋈

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML Text	Output
<b>My symbol: &#x22C8;</b>	My symbol: ⋈

On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22C8 to place the ⋈ symbol on your canvas. For example:

JavaScript Text
const x = "0x"+"E9" ctx.fillText(String.fromCodePoint(x), 5, 5);
Output
⋈

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ⋈ is U+22C8. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:

Type	Output
22C8 [Hold down Alt] [Press x]	⋈ (The 22C8 turns into ⋈. Note that you can omit any leading zeros.)

In JavaScript, the syntax is \uXXXX. So, our example would be \u22C8. (Note that the format is 4 hexadecimal characters.)

JavaScript Text	Output
let str = "\u22C8" document.write("My symbol: " + str)	My symbol: ⋈

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