What is the difference between two sets?

The difference between two sets is the set of elements that are in one set but not the other.

In set theory, the difference between two sets \(A\) and \(B\) is written as \(A - B\) or \(A \setminus B\). This operation results in a new set containing all the elements that are in set \(A\) but not in set \(B\). For example, if \(A = \{1, 2, 3, 4\}\) and \(B = \{3, 4, 5, 6\}\), then \(A - B = \{1, 2\}\) because 1 and 2 are in \(A\) but not in \(B\).

To understand this better, imagine you have two groups of friends. Group \(A\) includes friends who like football, and Group \(B\) includes friends who like basketball. If you want to find out which of your friends like only football and not basketball, you would look at the difference between Group \(A\) and Group \(B\). This helps you identify the friends who are unique to the football group.

In mathematical terms, the difference operation is useful for various applications, such as solving problems involving sets, analysing data, and understanding relationships between different groups. It is a fundamental concept in set theory and is often used in conjunction with other set operations like union and intersection to solve more complex problems.

Remember, the order in which you subtract sets matters. \(A - B\) is not the same as \(B - A\). Using our previous example, \(B - A = \{5, 6\}\), which are the elements in \(B\) but not in \(A\). This distinction is crucial when working with sets to ensure accurate results.