Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A\) and \(B\) be two sets then \((A \cup B)^{\prime} \cup\left(A^{\prime} \cap B\right)\) is equal to (A) \(B^{\prime}\) (B) \(B\) (C) \(A\) (D) \(A^{\prime}\)

Step-by-Step Solution

Answer

The expression simplifies to \(A^{\prime}\), so the answer is (D) \(A^{\prime}\).

Step 1: Understand the Expression

The given expression is \((A \cup B)^{\prime} \cup (A^{\prime} \cap B)\). This involves three operations: union (\(\cup\)), intersection (\(\cap\)), and complement (\(\prime\)). Our goal is to simplify this expression.

Step 2: Apply De Morgan's Law

First, apply De Morgan's Law to simplify \((A \cup B)^{\prime}\). De Morgan's Law states that \((A \cup B)^{\prime} = A^{\prime} \cap B^{\prime}\). So now, our expression becomes \((A^{\prime} \cap B^{\prime}) \cup (A^{\prime} \cap B)\).

Step 3: Factor Out the Common Term

Notice that both parts of the expression have \(A^{\prime}\). We can factor out \(A^{\prime}\) to simplify: \((A^{\prime} \cap B^{\prime}) \cup (A^{\prime} \cap B) = A^{\prime} \cap (B^{\prime} \cup B)\).

Step 4: Use the Complementary Law

According to the Complementary Law, \(B^{\prime} \cup B = U\), where \(U\) is the universal set. So now, \(A^{\prime} \cap (B^{\prime} \cup B) = A^{\prime} \cap U\) becomes just \(A^{\prime}\).

Step 5: Conclusion

The simplified expression is \(A^{\prime}\). Therefore, the correct answer is (D) \(A^{\prime}\).

Key Concepts

De Morgan's Law

De Morgan's Laws provide a fundamental principle in set theory that describes how the complement of a union or intersection of sets can be expressed. In particular, they offer a way to transform complex set expressions into simpler ones, making them easier to work with.

There are two main statements of De Morgan's Laws:

• The complement of the union of two sets is the intersection of their complements: \( (A \cup B)' = A' \cap B' \).
• The complement of the intersection of two sets is the union of their complements: \( (A \cap B)' = A' \cup B' \).

To apply De Morgan's Law, look for expressions involving the complement of a union or intersection. This can simplify complex logical statements in mathematics, computer science, and logic. In our exercise, we used De Morgan's Law to transform \( (A \cup B)' \) into \( A' \cap B' \), paving the way for further simplification.

Union and Intersection of Sets

In set theory, the union and intersection are ways to combine different sets to form new ones. Understanding these operations is crucial for solving problems related to sets.

- **Union (\( \cup \))**: The union of two sets \( A \) and \( B \) is a set containing all the elements that are in \( A \), in \( B \), or in both. It's like putting all the members of \( A \) and \( B \) together without duplicates. Mathematically expressed as \(A \cup B = \{ x : x \in A \text{ or } x \in B \} \).

- **Intersection (\( \cap \))**: The intersection of two sets \( A \) and \( B \) is a set containing only the elements that are common to both sets. They have to be in both sets \( A \) and \( B \). Expressed mathematically as \( A \cap B = \{ x : x \in A \text{ and } x \in B \} \).

These operations allow sets to be combined in various logical ways. In the original problem, you see these operations harmonizing with the complement to enable the simplification using rules like De Morgan's and the Complementary Law.

Complementary Law

The Complementary Law in set theory refers to principles that involve the complement of a set within a universe of discourse. This law simplifies expressions by determining how a set and its complement interact.

For any set \( B \), when you take the union of \( B \) and its complement \( B' \), you get the universal set \( U \), as \( B \cup B' = U \). Likewise, the intersection of \( B \) with its complement is the empty set: \( B \cap B' = \emptyset \).

In practical terms, these laws imply that a set and its complement together cover the entire universe of possible elements (union), while their overlap is nonexistent (intersection).

In the given solution, we used the Complementary Law to simplify \( A' \cap (B' \cup B) \) to just \( A' \cap U \). Here, since \( A' \) intersected with the universal set \( U \) yields just \( A' \), it provided the answer to our problem with a clear connection between these central set theory principles.