Question

Look at the following code: \\[\begin{array}{l}\text { set1 }=\operatorname{set}([1,2,3,4]) \\ \text { set } 2=\operatorname{set}([2,3])\end{array}\\] Which of the sets is a subset of the other? Which of the sets is a superset of the other?

Short answer

Answer: Set1 is a superset of set2, and set2 is a subset of set1.

Step-by-step solution

Understand the definition of subsets and supersets

A set A is a subset of set B if every element of A is also an element of B. Additionally, set B is a superset of set A if every element of A is also an element of B. In other words, if all elements of the first set are also in the second set, the first set is a subset of the second and the second is a superset of the first.

Identify the elements of set1 and set2

Let's identify the elements of each set to make comparisons easier. The given code shows that set1 has elements: 1, 2, 3, and 4, and set2 has elements: 2 and 3. Write these down for quick reference.

Compare the elements of set1 and set2

Now, compare the elements of the sets to determine if set1 is a subset or superset of set2: - set1 contains the elements 1, 2, 3, and 4 - set2 contains the elements 2, and 3 Since all of the elements in set2 (2 and 3) are also present in set1, we can say that set2 is a subset of set1. Furthermore, since set1 contains all of the elements of set2, set1 is a superset of set2.

Conclusion

After comparing the elements of set1 and set2, we can conclude that set1 is a superset of set2, and set2 is a subset of set1.

Key concepts

Set Theory

At the heart of mathematics, we find set theory, an area that deals with the collection of distinct objects, known as sets, and their relationship to each other. These objects, be it numbers, symbols, or even other sets, are called elements.

Understanding set theory is crucial because it serves as the foundation for various mathematical disciplines such as algebra, calculus, and probability. It also plays a pivotal role in logic and computer science. A set is described by its elements, and the order of these elements does not matter. Two fundamental properties used to describe the relationship between sets are 'subsets' and 'supersets'.

• A subset is a set whose elements are all contained in another set. If set A is a subset of set B, then everything in A is also in B.
• A superset is the converse; set B contains all elements of set A, plus possibly more.

These relationships are symbolized as A ⊆ B for subsets and B ⊇ A for supersets.

Programming with Python

Python is a versatile, high-level programming language that makes it easy to manipulate sets, thanks to its built-in set data type. In Python, sets are defined much like mathematical sets and provide a collection of unordered elements with no duplicates.

Understanding how to work with sets in Python can be extremely useful when solving a variety of problems, such as removing duplicates from a list, testing for membership of an element, or performing mathematical set operations like union, intersection, and difference. The code in the exercise demonstrates how to define two sets using list data and how to manipulate them using Python’s set operations.

• set() is a function that creates a set out of a list.
• issubset() and issuperset() methods can be used to check subset and superset relationships, respectively.

Python directly parallels the fundamental concepts of set theory, making it an excellent language for exploring these mathematical concepts.

Mathematical Sets Concepts

Diving into mathematical sets concepts can seem daunting at first, but understanding the basics of sets is quite straightforward. We deal with sets in our everyday life, from a collection of books on a shelf to the diverse species in a biological ecosystem. Mathematics formalizes this concept so we can apply it systematically.

In the context of the given exercise, we can see an application of mathematical sets concepts in a Python programming environment. The exercise requires comprehending the nature of subsets and supersets, a significant aspect of mathematical set theory, and applying this understanding to determine the relationship between two specific sets.

Correctly identifying subsets and supersets, as illustrated in the solution, is not only crucial in academic settings but can also translate to practical applications, such as database querying and logic programming. In these cases, the principles originate in mathematical set theory but are applied within the scope of the problem at hand.