Problem 4 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Set \(A\) and \(B\) have 3 and 6 elements respectively. What can be the minimum number of elements in \(A \cup B\) ? (A) 18 (B) 9 (C) 6 (D) 3

Step-by-Step Solution

Answer

The minimum number of elements in \( A \cup B \) is 6.

Step 1: Understand the Union Concept

The union of two sets, denoted as \( A \cup B \), includes all unique elements that are in either set \( A \), set \( B \), or in both.

Step 2: Analyze the Element Count

Set \( A \) has 3 elements, and set \( B \) has 6 elements. To find the minimum number of elements in \( A \cup B \), we consider the possibility that all elements of set \( A \) are also in set \( B \).

Step 3: Calculate Minimum Union

If all 3 elements of \( A \) are already part of \( B \), then the unique elements in \( A \cup B \) come entirely from \( B \). This means the minimum number of elements in \( A \cup B \) would be the number of elements in \( B \), which is 6.

Key Concepts

Union of Sets

The union of sets is a fundamental concept in set theory. When you take the union of two sets, you're essentially combining all elements from both sets into a new set. However, it's not just a simple addition of elements. The union focuses on the uniqueness of elements.When we write the union of two sets, say set \( A \) and set \( B \), we denote it as \( A \cup B \). This new set contains:

• All elements that are in set \( A \)
• All elements that are in set \( B \)
• Without any repeated elements

So, every element is included once even if it appears in both sets. This is why the union is crucial when dealing with overlapping sets.
In the example above, if all elements of set \( A \) are already contained in set \( B \), then the union doesn’t add any new elements from set \( A \), effectively only consisting of the elements in set \( B \). This principle helps solve problems like determining the minimum size of a set union.

Minimum Elements

The idea of minimum elements often plays a role in set theory problems, especially when understanding the union of sets. Minimum elements refer to the fewest unique elements possible in a union of two or more sets.To find the minimum number of elements in \( A \cup B \), one must look at the overlap between the sets. This involves checking if some elements in one set are already present in the other set.
In the specific scenario, we see that set \( A \) might be entirely contained within set \( B \). If each element of \( A \) is also an element of \( B \), this means no new elements are added when taking the union. Hence, the minimum number of elements is the number of elements in the larger set or set with no outside contributions, which is 6.
This concept of minimum elements is essential for efficiently solving problems related to set unions.

Element Count in Sets

Counting the number of elements in sets is a straightforward but critical step in set theory, especially when working with unions. Knowing how to analyze the size of the union in terms of the counts of the individual sets is essential. In the problem, set \( A \) has 3 elements, and set \( B \) has 6 elements. The task is to find how many unique elements would be there after the union. You should always start by adding the number of elements in both sets and then subtracting any overlap—if there's an overlap. Typically, the formula can be expressed as:
\[|A \cup B| = |A| + |B| - |A \cap B|\]Where \(|A \cap B|\) is the number of elements that are common to both sets.
For the minimum scenario, where all elements of \(A\) also exist in \( B \), \(|A \cap B|\) equals |A|, which reinforces our understanding that the minimum elements in the union can be simply those of set \( B \), showing how overlap directly impacts the resulting union size. This equation helps visualize element relationships, enabling smarter set manipulations.