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Is a letter considered to be in alphabetical order with itself?

English Language & Usage Asked by Alexandros on December 23, 2020

We’ve come across a maths problem regarding car license plates. Their format is XXN where each X is a letter from A-D, and N is a digit. For example "AA0, AB1, CB9, etc"

One of the questions asked is to calculate what fraction of all possible plates have their letters "in alphabetical order" and my 10-year old asked me: "is a plate that repeats the same letter twice such as AA1 considered to be in alphabetical order"?

While I am inclined to say yes, the objection is that the alphabet never repeats the same letter twice, it is always increasing.

2 Answers

TLDR: In math AAB is ordered, it is alphabetically ordered, but it is not strictly ordered.

In math, the assumption is yes, and you need to specify if it is not the case. Alphabets are totally ordered sets

In math, the term "ordered" means the operator has three properties. The one we are interested in is connexity. This property is: a<=b or b<=a for all a,b. With respect to AAB: it is ordered.

If you change the connexity property, you say "strictly ordered". AAB is not strictly ordered. If you need to discriminate between the two, you can use "non-strictly ordered" to refer to the former.

There are two recognized ways to alphabetize strings of letters. They differ in how they handle strings with different numbers of letters. But since the length is specified here, it doesn't matter, both will yield the same result.

The difficulty with these terms is that they are defined in a very general manner. The definitions are hard to understand at first glance, and mathematicians are loath to provide examples that lose generality.

Correct answer by Phil Sweet on December 23, 2020

Your maths problem restricts letters to a-d, so this is clearly not a problem about English language. To answer the question correctly, you need to ask the person who stated the question. As asked, the question is ambiguous.

A mathematician would have called it either “in increasing alphabetical order” or “in non-decreasing alphabetical order”. The letters AA would definitely not be “in increasing alphabetical order”, but would definitely be “in non-decreasing alphabetical order”.

In general, the question would be very country dependent. German license plates for example are not restricted to the letters a-z. There are 1 to 3 initial letters, followed by a hyphen and another letter, followed by digits but not 0 as the first digit. However, only about 700 of the 1-3 letter combinations are allowed.

In the U.K. on the other hand, the rules have changed over time, and as a result sorting correctly is a lot more complicated.

Answered by gnasher729 on December 23, 2020

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