Basic operations on sequences (conventional notation)

In summary, the person is asking if there is an equivalent operator for set-difference when dealing with sequences instead of sets. They also inquire about converting sequences to sets and if there is a counterpart to the union operator for sequences. They mention using the operator "f" to return the elements of a sequence as a set without repetitions.
  • #1
azal
8
0
Hi All,

So here's my question:

Suppose we have two sets [itex]A[/itex] and [itex] B[/itex], then [itex]A \setminus B[/itex] 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 [itex] \setminus [/itex] to get the result I'm looking for.

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

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

Thanks so much for your help,

-A.
 
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  • #2
azal said:
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!)
 
  • #3
So, assume the operator [itex] f [/itex] takes a sequence, and returns its elements as a set (without repetitions). For example if [itex]\mathbf{a} = (1,1,2,1,3,2) [/itex] then [itex] f(\mathbf{a}) = \{1,2,3\} [/itex].

Now suppose we have a pair of sequences [itex] \mathbf {a} = (a_1,a_2,\cdots,a_n)[/itex] and [itex] \mathbf {b} = (b_1,b_2,\cdots,b_m)[/itex]. I want [itex]\mathbf {a}-\mathbf {b}:= f(\mathbf {a})\setminus f(\mathbf{b})[/itex].
 
  • #4
So in your example I want the operator to produce: [itex]\{1,2,3,4,5\} \setminus \{1,3,4,6\} = \{2,5\}.[/itex]
 
  • #5


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
 

What are sequences and how are they represented using conventional notation?

Sequences are a set of ordered elements that follow a specific pattern or rule. They can be represented using conventional notation, which typically involves listing the elements in curly brackets and separating them with commas.

What are the basic operations that can be performed on sequences using conventional notation?

The basic operations on sequences include finding the next term in the sequence, finding the value of a specific term, and finding the sum or product of all the terms. These operations can be performed using various mathematical formulas and techniques.

How can we determine the pattern or rule of a sequence using conventional notation?

To determine the pattern or rule of a sequence in conventional notation, we can examine the relationship between each term and its position in the sequence. This can be done by finding the differences between consecutive terms or by looking for a common ratio or factor between the terms.

What are some common examples of sequences that can be represented using conventional notation?

Some common examples of sequences that can be represented using conventional notation include arithmetic sequences (where the difference between consecutive terms is constant), geometric sequences (where there is a common ratio between consecutive terms), and Fibonacci sequences (where each term is the sum of the two previous terms).

How are sequences and conventional notation used in real-world applications?

Sequences and conventional notation are used in various real-world applications, such as in finance to calculate compound interest or in computer science to generate patterns and series. They are also used in mathematical models to represent natural phenomena and in programming to create algorithms and codes.

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