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

Question

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

Step-by-Step Solution

Answer

The expression is equal to \(A^{\prime}\), which corresponds to option (D).

Step 1: Understand the Problem

We need to determine the expression \((A \cup B)^\prime \cup(A^\prime \cap B)\)\. It involves set operations: union, intersection, and complement.

Step 2: Break Down the Expression

Analyze the part \((A \cup B)^\prime\)\. By De Morgan's laws, the complement of the union is \((A \cup B)^\prime = A^\prime \cap B^\prime\)\.

Step 3: Evaluate the Second Term

Now evaluate \(A^\prime \cap B\)\. This represents the elements that are in \(B\) but not in \(A\).

Step 4: Simplify the Union

Combine \(A^\prime \cap B^\prime\) and \(A^\prime \cap B\) with a union: \( (A^\prime \cap B^\prime) \cup (A^\prime \cap B) \). Factor \(A^\prime\): = A^\prime \cap (B^\prime \cup B)\.

Step 5: Apply the Complement Property

Observe that \(B^\prime \cup B\) is the universal set since it includes all elements either in \(B\) or not in \(B\), which is the whole set.

Step 6: Conclusion from Simplification

Hence, \((A^\prime \cap (B^\prime \cup B)) = A^\prime\). Thus, \((A \cup B)^\prime \cup (A^\prime \cap B) = A^\prime\).

Key Concepts

Union of Sets

When discussing the union of sets in set theory, we refer to an operation that combines all elements from two sets. Consider sets \(A\) and \(B\). The union of \(A\) and \(B\), denoted as \(A \cup B\), consists of elements that belong to either \(A\), \(B\), or both.
For example, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), then their union, \(A \cup B\), equals \(\{1, 2, 3, 4, 5\}\).
The union operation is fundamental in set theory because it helps us understand and manipulate sets by bringing together all distinct elements from the involved sets.

• Used to combine elements from different sets.
• Includes elements from both sets without repetition.
• Represented in mathematics by the symbol \(\cup\).

Intersection of Sets

The intersection of sets is a concept where we focus on the common elements shared between sets. For two sets \(A\) and \(B\), the intersection, denoted \(A \cap B\), includes only the elements that are present in both \(A\) and \(B\).
If we take the same sets as before, \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), their intersection is \(A \cap B = \{3\}\). This demonstrates how only the elements present in both sets appear in the intersection.
The intersection is essential for understanding areas of overlap between different sets and is often used for logical operations and data analysis.

• Seeks the common elements within sets.
• Contains only elements found in every involved set.
• Denoted by the symbol \(\cap\).

De Morgan's Laws

De Morgan's Laws provide powerful rules for computing set complements involving unions and intersections. They show how the complement of a union relates to the intersection of complements and vice versa.
The laws state:

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

In our exercise, we used the first law to simplify the expression \((A \cup B)'\) to \(A' \cap B'\). This simplification plays a critical role in breaking down complex expressions into more manageable parts and aids in solving problems efficiently.
De Morgan's Laws are foundational in both set theory and logic because they provide clarity and simplify operations involving sets.