Basic operations on sequences (conventional notation)

[azal]

Hi All,

So here's my question:

Suppose we have two sets $A$ and $B$, then $A \setminus B$ denotes their set-difference.
Does there exist an equivalent operator for the case where A and B are not sets, but sequences?

Otherwise, is there an operator to convert a sequence into a set, removing the index, and all repetitions? In that case, I can take my sequences, convert them to their corresponding sets, and use $\setminus$ to get the result I'm looking for.

Also, what is the counterpart of $A \cup \{b\}$ for the case where $A$ is a sequence? is it $A \oplus b$?

I can't seem to find such conventions regarding sequences anywhere on the web ...

Thanks so much for your help,

-A.

 

[Stephen Tashi]

Does there exist an equivalent operator for the case where A and B are not sets, but sequences?

You aren't being specific enough abou what operation you want. For example,
if A = 1,2,3,4,5 and B = 1,3,3,4,6 what do you want "the equivalent operator" to do? Produce the sequence 1-1,2-3,3-3,4-4,5-6 = 0,-1,0,0,-1 ? Or produce the sequence 2,3,5,6 ? Or produce the sequence 2,6 ?

Some authors use the notation A - B to mean term-by-term subtraction. For more elaborate operations, I don't think there is any standard notation. If you are writing a paper on this specialized subject, look in the related literature and see what people have invented. (And don't feel obligated to use it!)

 

[azal]

So, assume the operator $f$ takes a sequence, and returns its elements as a set (without repetitions). For example if $\mathbf{a} = (1,1,2,1,3,2)$ then $f(\mathbf{a}) = \{1,2,3\}$.

Now suppose we have a pair of sequences $\mathbf {a} = (a_1,a_2,\cdots,a_n)$ and $\mathbf {b} = (b_1,b_2,\cdots,b_m)$. I want $\mathbf {a}-\mathbf {b}:= f(\mathbf {a})\setminus f(\mathbf{b})$.

 

[azal]

So in your example I want the operator to produce: $\{1,2,3,4,5\} \setminus \{1,3,4,6\} = \{2,5\}.$

 

[CellCoree]

Hello A,

Thank you for your question. It is true that there are some differences in notation and operations between sets and sequences. In general, sequences are considered to be ordered lists of elements, while sets are considered to be unordered collections of distinct elements.

To answer your first question, there is not a direct equivalent operator for set-difference when dealing with sequences. However, you can still achieve a similar result by using a combination of operations. For example, if you have two sequences A and B, you can first convert them into sets by removing any duplicate elements and then use the set-difference operator, as you mentioned in your question. Alternatively, you can also use a loop or filter function to remove any elements in A that are also in B, depending on the programming language or software you are using.

As for converting a sequence into a set, there is not a specific operator for this. However, you can achieve this by simply removing the indices and duplicate elements from the sequence. This can be done using a loop or filter function, as mentioned before.

In terms of the counterpart of A \cup \{b\} for sequences, there is not a universally agreed upon operator for this. Some sources use A \oplus b, as you mentioned, while others may use A \circ b or A \cdot b. It is important to note that the use of these operators may vary depending on the context and the specific field of study.

I hope this helps clarify the conventions surrounding sequences and their operations. Keep in mind that these conventions may vary and it is always important to check with your specific field or context for the most accurate and appropriate notation and operations.

Best of luck with your research,

, Scientist