Question

After the following code executes, what elements will be members of set3? $$\begin{array}{l}\text { set1 }=\operatorname{set}\left(\left[' a^{\prime}, \quad^{\prime} b^{\prime},\quad^{\prime}c^{\prime}\right]\right) \\ \text { set } 2=\operatorname{set}\left(\left[' b^{\prime}, \quad^{\prime} c^{\prime}, \quad^{\prime} d^{\prime}\right]\right) \\\\\text { set } 3=\text { set1. symmetric }_{-} \text {difference (set2) }\end{array}$$

Short answer

Answer: The elements that are present in set3 after performing a symmetric difference operation between set1 and set2 are {'a', 'd'}.

Step-by-step solution

Break down the sets

First, write down the given sets, as we will need them for the next step: $$\text { set1 } = \operatorname{set}\left(\left['a', \quad ' b', \quad 'c'\right]\right)$$ $$\text { set2 } =\operatorname{set}\left(\left[' b', \quad ' c', \quad 'd'\right]\right)$$

Determine symmetric difference

Next, we will find the symmetric difference between the two sets. This includes elements that are unique to set1 and set2: - Unique elements in set1: {'a'} - Unique elements in set2: {'d'}

Combine the unique elements

For the final step, combine the unique elements from set1 and set2 to form the symmetric difference (set3): $$\text { set3 } = \text { set1.symmetric_difference(set2) } = \{ 'a', 'd' \}$$ Therefore, the elements in set3 after performing a symmetric difference between set1 and set2 are {'a', 'd'}.

Key concepts

Set Operations

Set operations are fundamental to the study of mathematics, particularly in areas involving statistics, probability, and computer science. These operations enable us to combine, relate, and analyze collections of objects, known as sets, through various mathematical procedures.

One of the most important operations is the symmetric difference, which finds elements that are in either of the two sets but not in both. To visualize this, imagine two circles overlapping; the symmetric difference would include only the areas in either circle that do not overlap.

Aside from symmetric difference, other common set operations include union (combining all unique elements from both sets), intersection (elements common to both sets), and difference (elements in the first set but not in the second). These fundamental operations allow us to work through a variety of problems in set theory and beyond, offering a toolkit for dealing with collections of objects in a systematic way.

Python Sets

In Python, sets are a data type similar to lists or dictionaries, used to store multiple items in a single variable. However, what makes sets unique is that they are unordered, mutable, and do not allow duplicate elements. They are ideal for when the order of items is unimportant, and you want to ensure all items are distinct.

Python provides a variety of built-in methods to perform set operations. For instance, the symmetric_difference() method, used in our example, returns a set that contains all items from both sets, but not the items present in both. Python sets are powerful tools in programming, particularly for tasks involving uniqueness of elements and set operations such as unions, intersections, and differences. Learning to use Python sets effectively can greatly simplify coding tasks that involve collection data types.

Unique Elements in Sets

The concept of uniqueness is central to understanding sets—the idea that each element in a set is distinct and appears only once. Unique elements are the cornerstone of many set operations, allowing us to perform comparative analysis between different sets.

In the context of symmetric difference, we exclusively focus on unique elements from each set to form a new set. As demonstrated in the step-by-step solution, we identified the unique elements 'a' from set1 and 'd' from set2. These elements are what constitute the symmetric difference because they are not shared by both sets.

Identifying unique elements is not just theoretical but has practical uses in areas such as database querying, where one may need to filter out duplicate records, or in algorithm design, where ensuring distinct values can lead to more efficient processing. Understanding how to manipulate and extract unique elements from sets is a valuable skill in both mathematics and computer science.