Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(P, Q\) and \(R\) are subsets of a set \(\mathrm{A}\), then \(R \times\left(P^{\prime} \cup Q^{\prime}\right)^{\prime}\) equals (A) \((R \times P) \cap(R \times Q)\) (B) \((R \times Q) \cap(R \times P)\) (C) \((R \times P) \cup(R \times Q)\) (D) None of these

Step-by-Step Solution

Answer

(A) \( (R \times P) \cap (R \times Q) \)

Step 1: Understand Notations and Operations

First, recall that for sets, the complement of a set \( P \), denoted \( P' \), is the set of all elements not in \( P \). The union \( P \cup Q \) is the set of elements in either \( P \) or \( Q \) or both. The Cartesian product \( R \times S \) results in ordered pairs where the first element is from \( R \) and the second is from \( S \).

Step 2: Use De Morgan's Laws

According to De Morgan's laws, the complement of a union is the intersection of the complements: \((P' \cup Q')' = P \cap Q\). Hence, \((P' \cup Q')'\) simplifies to \( P \cap Q \).

Step 3: Simplify the Expression

Substitute the result from Step 2 into the given expression: \[ R \times (P' \cup Q')' = R \times (P \cap Q) \]This is the Cartesian product between set \( R \) and the intersection of sets \( P \) and \( Q \).

Step 4: Apply Cartesian Product Distribution

The Cartesian product distributes over intersections as follows:\[ R \times (P \cap Q) = (R \times P) \cap (R \times Q) \]This tells us that the given expression simplifies to the intersection of the Cartesian products \( R \times P \) and \( R \times Q \).

Step 5: Compare With Options

Compare the simplified expression \( (R \times P) \cap (R \times Q) \) with the given options. This matches with option (A).

Key Concepts

Cartesian product

In set theory, the Cartesian product is a fundamental concept that allows us to combine elements from two sets to form ordered pairs. Given two sets, say, \( R \) and \( S \), the Cartesian product \( R \times S \) is the set of all possible ordered pairs where the first element is from \( R \) and the second element is from \( S \). For example, if \( R = \{1, 2\} \) and \( S = \{a, b\} \), then the Cartesian product \( R \times S = \{(1, a), (1, b), (2, a), (2, b)\} \). This essentially lists all possible ways we can pair an element from \( R \) with an element from \( S \).

• The order of the elements matters in Cartesian products, so \((a, b)\) is different from \((b, a)\).
• It is important to realize that if one of the sets is empty, the Cartesian product will also be empty, as there are no elements to pair.

De Morgan's laws

De Morgan's laws are essential tools in set theory and logic. These laws provide a way to connect the operations of union and intersection with complements. Let's look at the two main statements of De Morgan's laws:

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

In the context of the exercise, these laws are used to simplify expressions involving complements and unions. Specifically, they allow us to transform \((P' \cup Q')'\) into \( P \cap Q \). This conversion simplifies complex set expressions by applying logical negations more straightforwardly.

Set complement

The concept of a set complement is key to understanding how set differences are represented. If you have a universal set \( U \) and a subset \( A \), the complement of \( A \), denoted by \( A' \), consists of all the elements in \( U \) that are not in \( A \). Think of the set complement like flipping a switch where all elements that are in \( A \) are turned off, thus highlighting only those in \( U \) but not in \( A \). For example, if \( U = \{1, 2, 3, 4\} \) and \( A = \{2, 3\} \), then \( A' = \{1, 4\} \).

• Complement sets change in relation to the universal set, so what is part of \( A' \) depends entirely on what is defined as \( U \).
• This concept is especially useful when using De Morgan's laws to manipulate and simplify set expressions.

Set intersection

Understanding set intersection is critical in set theory because it defines what is common between sets. The intersection of two sets \( A \) and \( B \), denoted by \( A \cap B \), includes all elements that are both in \( A \) and \( B \).For example, if \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), then \( A \cap B = \{2, 3\} \). Here, \( 2 \) and \( 3 \) are present in both sets, making them members of the intersection.

• Intersections are often used in various operations to find shared properties or values.
• In the exercise solution, intersection is used significantly when simplifying expressions using De Morgan's laws and with Cartesian product distribution.

Breaking a complex problem into common elements helps when evaluating and comparing different set operations.