## The Mathematical Symbol "Multiset Union (⊎)"

The ⊎ Symbol in Mathematics: Multiset Union

Mathematics possesses a rich tapestry of symbols that enable concise expression of multifaceted concepts. The ⊎ symbol, representing the "Multiset Union", is one such symbol that holds significance in set theory. This article aims to shed light on the primary applications and relevance of the ⊎ symbol, enhanced by relevant examples.

## Usage

The ⊎ symbol is distinct from the regular union symbol used in set theory. It denotes the union of multisets. While traditional sets don't account for multiple occurrences of elements, multisets do. The ⊎ symbol combines multisets, counting repeated occurrences of elements from both multisets.

## Examples

• Example 1: Consider two multisets A = {1,1,2} and B = {1,2,2}. Their multiset union can be represented as:
A ⊎ B = {1,1,1,2,2,2}
Note that the ⊎ symbol has effectively combined the occurrences of each element from both A and B.
• Example 2: For multisets X = {a,a,b} and Y = {a,b,b,c}, the multiset union is:
X ⊎ Y = {a,a,a,b,b,b,c}
Here, the ⊎ ensures the aggregation of repeated occurrences of elements from both X and Y.

In summary, the ⊎ symbol offers a nuanced way to represent unions in the realm of multisets, where multiple occurrences of elements are paramount. Its introduction enables mathematicians to explore and operate on multisets with precision and clarity.

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## Codes for the ⊎ Symbol

 The Symbol ⊎ Alt Code Alt 8846 HTML Code ⊎ HTML Entity ⊎ CSS Code \228E Hex Code ⊎ Unicode U+228E

## 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 8846. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

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

HTML TextOutput
<b>My symbol: &#8846;</b>My symbol: ⊎

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

HTML TextOutput
<b>My symbol: &uplus;</b>My symbol: ⊎

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

CSS and HTML TextOutput
<style>
span:after {
content: "\228E";}
</style>
<span>My symbol:</span>
My symbol: ⊎

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

HTML TextOutput
<b>My symbol: &#x228E;</b>My symbol: ⊎
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x228E 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+228E. 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:
TypeOutput
228E
[Hold down Alt]
[Press x]

(The 228E turns into ⊎. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u228E. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u228E"
document.write("My symbol: " + str)
My symbol: ⊎