Problem 140 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(X=\left\\{4^{n}-3 n-1: n \in N\right\\}\) and \(Y=\\{9(n-1): n \in N\\}\) where \(\mathrm{N}\) is the set of natural numbers, then the set \(X \cup Y\) is equal to (A) \(N\) (B) \(Y-X\) (C) \(X\) (D) \(Y\)

Step-by-Step Solution

Answer

(D) Y

Step 1: Understanding Sets X and Y

Set \(X\) is defined as \(X = \{4^n - 3n - 1 : n \in \mathbb{N}\}\). This means each element of \(X\) is calculated by substituting natural numbers into the expression \(4^n - 3n - 1\). Similarly, set \(Y\) is defined as \(Y = \{9(n-1) : n \in \mathbb{N}\}\), meaning each element is obtained by substituting natural numbers into \(9(n-1)\). Both elements are derived from substitutions with natural numbers.

Step 2: Exploring Values in X

Let’s find the first few values of \(X\) by substituting natural numbers.For \(n=1\), \(4^1 - 3 \cdot 1 - 1 = 4 - 3 - 1 = 0\).For \(n=2\), \(4^2 - 3 \cdot 2 - 1 = 16 - 6 - 1 = 9\).Continue in this way to establish a pattern to understand the numbers in \(X\).

Step 3: Exploring Values in Y

Similarly, let's substitute natural numbers in \(Y\).For \(n=1\), \(9(1-1) = 9 \cdot 0 = 0\).For \(n=2\), \(9(2-1) = 9 \cdot 1 = 9\).Continue this for the next few natural numbers till we observe a pattern.

Step 4: Comparing Set Elements

Compare the number sequences of the first few elements from the calculations of \(X\) and \(Y\). Notice that for the beginning values, both sets contain the same elements \(0, 9, ...\), showing that \(X\) and \(Y\) have common elements.

Step 5: Determine the Union of Sets X and Y

The union \(X \cup Y\) consists of all the elements present in either set \(X\), set \(Y\), or both. From previous steps, every element of \(Y\) is also in \(X\), hence their union does not add new elements beyond set \(X\). Thus, \(X \cup Y = Y\).

Step 6: Final Comparison and Answer Selection

Upon detailing iterations and understanding sets \(X\) and \(Y\), the elements indicate that \(X \cup Y\) simplifies to set \(Y\) as it contains exactly the same elements as the derived pattern from \(X\). Hence, the set \(X \cup Y\) equals set \(Y\) alone, suggesting the answer is \(D: Y\).

Key Concepts

Natural Numbers

Natural numbers are the simplest set of numbers we often start with in mathematics, typically consisting of the positive integers beginning from 1 (like 1, 2, 3, and so on). They are used for counting and ordering in everyday life. These numbers have no fractions or decimals and are infinite, meaning they never end.
In mathematical terms, natural numbers are usually denoted by the symbol \(\mathbb{N}\). The concept of natural numbers provides a fundamental building block for various areas in mathematics, including algebra and set theory.
In set definitions like the ones given in the exercise, natural numbers are often used to determine initial elements or to function as parameters in expressions. Understanding these is critical, as they form the basis for exploring more complex mathematical concepts.

Set Theory

Set theory is a branch of mathematical logic that deals with collections of objects, known as sets. In the provided exercise, sets are crucial as they define groups of numbers created by certain expressions.
A set is any well-defined collection of distinct objects, which can include numbers, symbols, or even other sets. For example, the set \(X\) given by \(X = \{4^n - 3n - 1 : n \in \mathbb{N}\}\) means that every element in the set is determined by substituting natural numbers into the expression \(4^n - 3n - 1\).
The operation of union, denoted as \(X \cup Y\), is essential in set theory. It refers to a set that includes all elements that belong to either \(X\), \(Y\), or both.

• If any element is in either one or both sets, it becomes a part of the union.
• The result in the exercise shows how all elements of set \(Y\) are contained within set \(X\), making their union simply \(Y\).

Understanding set theory helps in organizing and explaining the relationships and operations like unions, intersections, and differences of sets.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. In the context of the given exercise, expressions like \(4^n - 3n - 1\) and \(9(n-1)\) are crucial to generating the elements of the sets \(X\) and \(Y\), respectively.
An algebraic expression includes combination of variables and constants that are combined using operations such as addition, subtraction, multiplication, and exponentiation. These expressions can represent a wide range of numbers based on the values substituted for the variables.

• In set \(X\), the expression generates numbers by calculating \(4^n\), subtracting \(3n\), and then further reducing by 1 for each natural number \(n\).
• In set \(Y\), the expression calculates for every natural number \(n\) by finding out \((n-1)\) first and then multiplying it by 9.

By exploring different values of \(n\), we can identify patterns or specific numbers present in each set. Understanding algebraic expressions allows us to manipulate and comprehend complex mathematical problems, which are key in approaching exercises involving sets and unions.