Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A=\\{x: x \in R,|x|<1\\}\) \(B=\\{x: x \in R,|x-1| \geq 1\\}\) and \(A \cup B=R-D\), then the set \(D\) is (A) \(\\{x: 1

Step-by-Step Solution

Answer

The set \(D\) is \(\{1\}\), so the answer is (D) None of these.

Step 1: Understanding Set A

The set \(A\) is defined as those real numbers \( x \) such that the absolute value of \( x \) is less than 1. This translates to the inequality: \(-1 < x < 1\). So, \(A = \{ x \mid -1 < x < 1 \} \).

Step 2: Understanding Set B

The set \(B\) is defined by the condition \(|x - 1| \geq 1\). This splits into two inequalities: \(x - 1 \geq 1\) or \(x - 1 \leq -1\), which simplify to \(x \geq 2\) or \(x \leq 0\). Thus, \(B = \{ x \mid x \geq 2 \} \cup \{ x \mid x \leq 0 \}\).

Step 3: Finding \(A \cup B\)

The union \(A \cup B\) includes all elements in either \(A\) or \(B\). From our earlier analysis, \(A = \{ x \mid -1 < x < 1 \}\) and \(B = \{ x \mid x \geq 2 \} \cup \{ x \mid x \leq 0 \}\). This means \(A \cup B = \{ x \mid x eq 1 \} \).

Step 4: Identifying \(D\) From \(A \cup B = \mathbb{R} - D\)

Since \(A \cup B = \{ x \mid x eq 1 \}\) and \(A \cup B = \mathbb{R} - D\), it follows that \(D\) consists of elements that are not included in \(A \cup B\). Therefore, \(D = \{1\}\).

Step 5: Comparing With Options

Among the given options, we check which one matches \(D = \{1\}\). Option (D) 'None of these' is the only option that correctly represents \(D = \{1\}\).

Key Concepts

Real Numbers

Real numbers are the set of numbers that include both rational and irrational numbers. They can be found on the number line and include negative numbers, zero, and positive numbers. Real numbers are crucial in mathematics because they include all the possible values that any quantity can have, especially when you are dealing with things like measurements, calculations, and various mathematical models.

Some key points about real numbers:

• **Rational Numbers:** Numbers that can be expressed as a fraction or ratio of two integers, such as \ \( \frac{3}{4} \ \) or 0.5.
• **Irrational Numbers:** Numbers that cannot be expressed as simple fractions, such as \ \( \pi \ \) or the square root of 2.
• Real numbers extend infinitely in both the positive and negative directions on the number line.

Union of Sets

In set theory, the union of two sets, say set \( A \) and set \( B \), is a set containing all the elements from both sets. This is represented as \( A \cup B \). The union includes every element that belongs to either \( A \), \( B \), or both.

When calculating the union, you are essentially combining the elements of both sets without repetition. For example, if \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), then \( A \cup B = \{1, 2, 3, 4, 5\} \).

For the solution in the exercise, the union of sets \( A \) and \( B \) was explored with real numbers where \( A = \{x: -1 < x < 1\} \) and \( B = \{x: x \geq 2\} \cup \{x: x \leq 0\} \). When combined, all relevant elements that satisfy either condition were included, forming a complete set of real numbers excluding one specific value, in our case.

Set Complement

The complement of a set refers to elements not in the set, often compared against a universal set \( U \). For example, if \( U \) includes all real numbers, then the complement of set \( A \), denoted as \( A' \) or \( \bar{A} \), would consist of all real numbers not in set \( A \).

In this exercise, we used the complement concept to find the set \( D \). Since \( A \cup B = \mathbb{R} - D \), this implies that \( D \) consists of all elements that are not in \( A \cup B \). This is like asking what parts of the universal set \( \mathbb{R} \) are missing from \( A \cup B \).

Understanding complements is vital in logic and probability, where it's important to consider not just the elements that are present, but also what has been excluded.

Inequalities in Sets

Inequalities in sets help define the boundaries of the elements that can be included in a set. These inequalities often involve comparing elements using less than \( < \), less than or equal to \( \leq \), greater than \( > \), and greater than or equal to \( \geq \) symbols.

For instance, when set \( A \) is defined as \( \{x: -1 < x < 1\} \), it includes all real numbers between -1 and 1. This uses the concept of inequalities to specify that -1 and 1 are not included but everything in between is. Similarly, set \( B \) is split with inequalities into two parts: \( \{x: x \geq 2\} \) and \( \{x: x \leq 0\} \).

Working with inequalities allows us to identify precisely which elements belong in a set, which is particularly valuable when sets have overlapping or exclusive elements. Understanding how to interpret these boundaries is foundational for grasping more complex mathematical problems.