Problem 14 - A Complete Resource Book in Mathematics for JEE Main

Question

Consider the following relations: (1) \(A-B=A-(A \cap B)\) (2) \(A=(A \cap B) \cup(A-B)\) (3) \(A-(B \cup C)=(A-B) \cup(A-C)\) Which of these is/are correct? (A) 1 and 3 (B) 2 only (C) 2 and 3 (D) 1 and 2

Step-by-Step Solution

Answer

(D) 1 and 2

Step 1: Analyze Relation (1)

We examine the relation \(A-B=A-(A \cap B)\). The left side, \(A-B\), means elements in \(A\) but not in \(B\). The right side, \(A-(A \cap B)\), also represents elements in \(A\) but not where \(A\) and \(B\) intersect. Both sides describe the same set, making relation (1) correct.

Step 2: Evaluate Relation (2)

We analyze \(A=(A \cap B) \cup (A-B)\). The left side is just set \(A\). The right side combines all elements in \(A\) that are also in \(B\) (\(A \cap B\)) and all elements unique to \(A\) (\(A-B\)). This indeed reconstructs \(A\) as it combines both common and unique parts of \(A\) which confirms relation (2) as correct.

Step 3: Test Relation (3)

We assess \(A-(B \cup C)=(A-B) \cup (A-C)\). The left side is elements in \(A\) not in either \(B\) or \(C\). The right side, \((A-B) \cup (A-C)\), includes elements not in \(B\) or not in \(C\), but doesn't cover situations where an element may be in \(C\) but not in \(B\) or vice versa. Thus, this does not always include elements exclusive to \(A\) only because it counts repeated elements, making relation (3) incorrect.

Key Concepts

Relations in Set Theory

In set theory, a relation defines a way in which two sets interact or connect with each other. Relations include familiar operations like union, intersection, and set difference. Understanding these relations helps us solve problems by identifying how elements from different sets can be combined or separated.

In the problem, relation (1) explores set difference, represented by \( A - B \). This operation identifies elements in \( A \) that are not in \( B \). Relation (1) states \( A - B = A - (A \cap B) \). Both sides describe the same outcome: elements in \( A \), excluding those shared with \( B \). Thus, this relation is valid.

Understanding relations helps in visualizing how combining or excluding sets affects the elements. By evaluating each expression, you can verify the truth behind these relations, enhancing your problem-solving skills.

Set Operations

Set operations allow us to manipulate and combine different sets for a variety of purposes in mathematics. Key operations include union, intersection, and set difference. These operations simplify complex problems by breaking them down into understandable parts.

Let's analyze the set operations involved in the problem. The union, represented by \( A \cup B \) or \( A \cup C \), combines elements from both sets, while intersection, denoted by \( A \cap B \), finds common elements between two sets. The set difference, \( A - B \), keeps elements in \( A \) but removes those found in \( B \).

In relation (2), we examined \( A = (A \cap B) \cup (A - B) \). The operation correctly reconstructs set \( A \) by capturing both shared and unique elements, affirming its accuracy. Understanding these operations aids in solving a wide range of mathematical issues efficiently.

Mathematical Problem Solving

Mathematical problem solving often involves applying concepts like set operations and relations strategically. These core features of set theory simplify understanding how different sets interact, making complex problems more manageable.

In solving the given exercise, problem-solving involves diligently evaluating each provided relation against known set theories. Relation (3) proposes \( A - (B \cup C) = (A - B) \cup (A - C) \). However, misinterpretation or oversight can lead to inaccuracies. Recognizing when the logic fails helps identify incorrect assumptions, as seen in relation (3).

Mathematical problem solving isn't just about finding the right answer; it's about learning from misconceptions, refining methods, and validating results using logical reasoning and theoretical principles. With practice, these skills enhance proficiency in tackling various mathematical challenges.