Question

Prove that if \(\mathrm{A} \cup \mathrm{B}=A\) and \(\mathrm{A} \mathrm{n} \mathrm{B}=\mathrm{A}\), then \(\mathrm{A}=\mathrm{B}\).

Short answer

By analyzing the set relations, we determine that \(A = B\).

Step-by-step solution

Understanding the Given Equations

We are given two conditions: \( A \cup B = A \) and \( A \cap B = A \). This means that the union of sets \(A\) and \(B\) is the set \(A\) itself, and the intersection of sets \(A\) and \(B\) is also \(A\).

Analyzing the Union Condition

The first condition \( A \cup B = A \) implies that every element of \(B\) is already in \(A\). Thus, \(B\) does not add any new elements to \(A\) when we perform the union, which essentially means \(B \subseteq A\).

Analyzing the Intersection Condition

The second condition \( A \cap B = A \) implies that every element of \(A\) is also in \(B\), since the intersection of \(A\) and \(B\) contains all the elements of \(A\). This means \(A \subseteq B\).

Combining Subset Relations

From the above two analyses, we have both \( B \subseteq A \) and \( A \subseteq B \). These two subset relations imply that the sets \(A\) and \(B\) must be equal. Therefore, \(A = B\).

Key concepts

Union of Sets

In set theory, the union of two sets is a fundamental concept that helps in understanding the combination of elements. The union of sets \(A\) and \(B\), denoted as \(A \cup B\), includes every element that is in \(A\), or \(B\), or in both.

It's important to note that the union operation considers all unique elements, so duplicates are automatically resolved.

For example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \cup B = \{1, 2, 3, 4, 5\}\).

In the original exercise, the equation \(A \cup B = A\) suggests that \(B\) is a subset of \(A\), meaning that \(B\) does not bring in any new elements to the set \(A\). All elements of \(B\) are already included in \(A\).

Intersection of Sets

The intersection of two sets \(A\) and \(B\) is a set containing only the elements that occur in both \(A\) and \(B\). This is represented as \(A \cap B\).

When you perform an intersection, you effectively narrow down to the common elements shared between the two sets.

Continuing the previous example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then \(A \cap B = \{3\}\).

In the exercise, the equation \(A \cap B = A\) indicates that \(A\) contains all the elements necessary for the intersection to also be \(A\). In simpler terms, every element of \(A\) is present in \(B\), making \(A\) a subset of \(B\).

Subset Relation

A subset is a set whose elements are all contained within another set. In notation, if every element of set \(B\) is also in set \(A\), then \(B\) is a subset of \(A\), expressed as \(B \subseteq A\).

If both \(B \subseteq A\) and \(A \subseteq B\) hold true, then the sets are equal, i.e., \(A = B\). This happens because both sets must contain the exact same elements.

• The set \(\{1, 2\}\) is a subset of \(\{1, 2, 3, 4\}\) because every element in the first set is also in the second.
• However, \(\{3, 4\}\) is not a subset of \(\{1, 2\}\), since not all its elements are contained within the second.

In the exercise, the application of subset relations—\(A \subseteq B\) and \(B \subseteq A\)—leads us to conclude that \(A\) and \(B\) must be identical, or \(A = B\). This reveals that under certain conditions involving union and intersection, the concept of subsets helps us determine set equality.