Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(X=\\{1,2,3,4,5\\} .\) The number of different ordered pairs \((Y, Z)\) that can formed such that \(Y \subseteq X, Z \subseteq X\) and \(Y \cap Z\) is empty, is (A) \(5^{2}\) (B) \(3^{5}\) (C) \(2^{5}\) (D) \(5^{3}\)

Step-by-Step Solution

Answer

The number of different ordered pairs is 243 (option B: \(3^5\)).

Step 1: Understanding the Subsets

We need to find pairs of subsets \((Y, Z)\) such that \(Y \subseteq X\), \(Z \subseteq X\), and \(Y \cap Z = \emptyset\). This means no element is included in both subsets simultaneously.

Step 2: Assigning Elements in \(X\)

Each element in set \(X\) can be placed in one of three categories for the pair \((Y, Z)\): the element can either be in \(Y\) only, in \(Z\) only, or not in either subset \(Y\) or \(Z\). Importantly, it cannot be in both \(Y\) and \(Z\) at the same time.

Step 3: Calculating Choices for Each Element

For each of the 5 elements in \(X\), we have exactly 3 choices: (1) include the element in \(Y\), (2) include it in \(Z\), or (3) exclude it from both \(Y\) and \(Z\).

Step 4: Computing Total Number of Combinations

Since the choices for each element are independent of each other and each element of \(X\) has 3 choices, the total number of possible ordered pairs \((Y, Z)\) is given by the product of these choices: \(3^5\).

Step 5: Final Calculation

Compute \(3^5 = 243\). Therefore, there are 243 different ordered pairs \((Y, Z)\) that satisfy the given conditions.

Key Concepts

Subsets

A subset is a set formed by taking some or none of the elements from a larger set. With the original set given as \(X = \{1,2,3,4,5\}\), any combination of these elements creates a possible subset. We represent this with \(Y \subseteq X\) or \(Z \subseteq X\), meaning \(Y\) and \(Z\) are subsets of \(X\).
To better understand subsets, think of each element in \(X\) having the choice to be included in or excluded from \(Y\) and \(Z\). The challenge is to form all possible combinations of these choices, respecting the condition that \(Y \cap Z = \emptyset\). This rule implies that no element can be chosen for both subsets \(Y\) and \(Z\) at the same time.

• Every subset, including the empty set and the set itself \(X\), is part of the power set of \(X\).
• The power set, in general, counts up to \(2^n\) subsets of a set with \(n\) elements.

Understanding these principles will aid in calculating the possible ordered pairs defined by the exercise guidelines.

Ordered Pairs

Ordered pairs combine two elements in a specific sequence, where order matters. This is crucial to remember: in the ordered pair \((Y, Z)\), \(Y\) comes first and \(Z\) second.
Unlike other combinations in mathematics, switching the order of elements would create a different ordered pair. This distinction is vital when constructing pairs of subsets in set theory. In this problem, the ordered pair is composed of two sets \(Y\) and \(Z\), each a subset of the main set \(X\).
Typically, the number of such ordered pairs will depend on how many ways we can assign elements to each subset while respecting constraints. In this case, \(Y \cap Z = \emptyset\) is our guiding restriction. This means no element is contained in both \(Y\) and \(Z\) simultaneously, ensuring the pair differs in how the same element is distributed.

• Ordered pairs are essential for understanding how data can be organized in math and computer science.
• They help structure input and output in many practical applications, from databases to algorithms.

Set Operations

Set operations focus on how different sets interact with each other through specific rules. Some fundamental operations involve union, intersection, and complement within the realm of set theory. Here, we're dealing with the concept of intersection \(Y \cap Z\), which must be empty.
This condition shapes how we can form our subsets of \(X\). Every element has three possible operations concerning the subsets \(Y\) and \(Z\): 1. It can be included in \(Y\) only.2. It can be included in \(Z\) only.3. It can be excluded from both \(Y\) and \(Z\).The intersection operation is crucial for determining overlap, or lack thereof. In this case, we want the overlap to be empty, guiding our assignment of elements to subsets.

• Employing set operations can simplify complex problems through formal logic and structure.
• Set theory is foundational in mathematics, laying groundwork for functions, relations, and probability.

Understanding these operations enables us to compute the way elements are distributed, ultimately leading to the solution where \(3^5\) combinations fulfill the conditions.