Rewriting Sets Using Set Comprehension and List Comprehension

Set comprehension is a powerful tool in both mathematics and computer programming for generating sets based on specific conditions. In this article, we will explore how to use set comprehension and list comprehension to rewrite a set, specifically the set defined as

{xy x ∈ {-3, -2, 1, 2, 3} and y ∈ {-1, 0, 1, 2}}

Understanding the Set

The given set {xy x ∈ {-3, -2, 1, 2, 3} and y ∈ {-1, 0, 1, 2}} represents a set of all possible sums of two numbers, where the first number (x) is from the set {-3, -2, 1, 2, 3} and the second number (y) is from the set {-1, 0, 1, 2}. In simple terms, we are looking to find all the sums of these pairs without repetition.

Rewriting the Set Manually

To understand the process, let's illustrate manually. We start by taking the first element of the set {x} and adding it to each element of the set {y}. We repeat this process for each element of the set {x} and identify the resulting sums:

x -3 -3 (-1) -4 -3 0 -3 -3 1 -2 -3 2 -1 x -2 -2 (-1) -3 -2 0 -2 -2 1 -1 -2 2 0 x 1 1 (-1) 0 1 0 1 1 1 2 1 2 3 x 2 2 (-1) 1 2 0 2 2 1 3 2 2 4 x 3 3 (-1) 2 3 0 3 3 1 4 3 2 5

From the above sums, we can see that the unique sums obtained are: -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. Therefore, the rewritten set is:

{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

Rewriting the Set Using Python and List Comprehension

Python is a popular language for mathematical and programming tasks due to its ease of use and strong support for set and list operations. Here's how you can implement this using list comprehension:

print(sorted([x   y for x in {-3, -2, 1, 2, 3} for y in {-1, 0, 1, 2}]))

This code generates all sums of the pairs and sorts the resulting list. The output is exactly the set of unique sums we found manually:

[-4, -3, -2, -1, 0, 1, 2, 3, 4, 5]

Conclusion

Using set and list comprehensions, both in mathematics and programming contexts, can greatly simplify the process of generating complex sets. By breaking down the problem and using the power of these tools, we can efficiently find the desired set of unique sums. This method not only ensures accuracy but also provides a clear, understandable process.