Problem 23 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(R\) be the real line. Consider the following subsets of the plane \(R \times R\). \(S=[(x, y): y=x+1\) and \(0

Step-by-Step Solution

Answer

(D) \( T \) is an equivalence relation on \( R \) but \( S \) is not.

Step 1: Evaluate Set S for Equivalence Properties

To determine if a set is an equivalence relation, it must satisfy reflexivity, symmetry, and transitivity. Set \( S \) is given as \([(x, y): y=x+1 \text{ and } 0

Step 2: Evaluate Set T for Equivalence Properties

Set \( T \) is defined as \([(x, y): x-y \text{ is an integer}]\). Let's check the conditions:- **Reflexivity**: For any \( x \) in \( R \), \( (x, x) \) is in \( T \) because \( x - x = 0 \) is an integer.- **Symmetry**: If \( (x, y) \in T \) means \( x - y \) is an integer, then \( y - x \) will also be an integer (since \( x - y = -(y-x) \)), so \( (y, x) \in T \).- **Transitivity**: If \( (x, y) \in T \) and \( (y, z) \in T \), then both \( x-y \) and \( y-z \) are integers. Thus, \( (x-z = (x-y) + (y-z)) \) is also an integer, so \( (x, z) \in T \).**Conclusion for T**: \( T \) is an equivalence relation.

Step 3: Identify the Correct Statement

From our evaluations above, we conclude: - \( S \) is not an equivalence relation because it does not satisfy reflexivity, symmetry, or transitivity.- \( T \) is an equivalence relation because it satisfies all three properties: reflexivity, symmetry, and transitivity.Thus, the correct statement is that \( T \) is an equivalence relation on \( R \) while \( S \) is not.

Key Concepts

Real Numbers

Real numbers form a set of numbers that include both rational numbers, like 2 or -5.5, and irrational numbers, such as \( \sqrt{2} \) and pi (\( \pi \)). They cover all points on the infinite number line, extending in both positive and negative directions. Real numbers can be classified into distinct categories:

• Natural Numbers: The counting numbers starting from 1, 2, 3, and so on.
• Whole Numbers: Natural numbers along with zero, so 0, 1, 2, 3, etc.
• Integers: All positive and negative whole numbers, including zero.
• Rational Numbers: Numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
• Irrational Numbers: Numbers that cannot be written as a simple fraction. These are often decimal numbers that do not terminate or repeat, like \( \sqrt{2} \) or \( \pi \).

Real numbers are essential for describing quantities that can have virtually any size or precision. They are vital in various mathematical fields, including calculus, algebra, and beyond.

Reflexivity

Reflexivity is one of the three critical properties an equivalence relation must satisfy. For a set to be reflexive, each element must relate to itself. Using the mathematical notation, for any element \( x \) in set \( R \), the pair \( (x, x) \) should exist in the relation. For example, considering set \( T = \{ (x, y): x-y \text{ is an integer} \} \), this property is satisfied because for any \( x \), \( x-x = 0 \) is indeed an integer. Thus, every real number is reflexive with itself in set \( T \). However, for set \( S = \{ (x, y): y = x + 1 \} \), reflexivity fails because \( x = x + 1 \) is never true for any real number \( x \). Therefore, this set does not satisfy the reflexivity condition. Reflexivity ensures that every item is in a self-loop when imagined as a vertex in a graph.

Symmetry

Symmetry requires that if an element \( x \) is related to \( y \), then \( y \) should also be related to \( x \). In simpler terms, if \( (x, y) \) is in the relation, \( (y, x) \) must also be part of it.In set \( T = \{ (x, y): x-y \text{ is an integer} \} \), this is satisfied. If \( x - y \) is an integer, then \( y - x \) is also an integer because it is just the negative of \( x - y \). Thus, symmetry holds true in \( T \).Contrarily, in set \( S = \{ (x, y): y = x + 1 \} \), symmetry fails. While \( y = x + 1 \) implies a particular direction from \( x \) to \( y \), reversing this statement results in \( x = y + 1 \), which doesn't hold true for the same pair, violating symmetry. Symmetry ensures a bidirectional relationship in pairs.

Transitivity

Transitivity is the property that if \( x \) is related to \( y \), and \( y \) is related to \( z \), then \( x \) should also be related to \( z \). It's like a chain reaction that connects elements across a set. In set \( T = \{ (x, y): x-y \text{ is an integer} \} \), transitivity holds. If \( x-y \) and \( y-z \) are both integers, then their sum \( x-z = (x-y) + (y-z) \) is also an integer, maintaining the transitive property.Conversely, in set \( S = \{ (x, y): y = x + 1 \} \), transitivity is absent. If \( y = x + 1 \) and \( z = y + 1 \), you get \( z = x + 2 \), not \( x + 1 \), which doesn’t fall back into set \( S \). This missing link shows that \( x \) and \( z \) aren't directly relatable through the same rule, thus, transitivity is not satisfied in set \( S \).

Previous

Problem 22

Next

Problem 24

Other exercises in this chapter

Problem 21

Let \(R\) be the set of real numbers. Statement 1: \(A=\\{(x, y) \in R \times R: y-x\) is an integer \(\\}\) is an equivalence relation of \(R\). Statement \(2:

View solution

Problem 22

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 empt

View solution

Problem 24

Let \(R=\\{(1,3),(4,2),(2,4),(2,3),(3,1)\\}\) be a relation on the set \(A=\\{1,2,3,4\\} .\) The relation \(R\) is \(\quad\) [2004] (A) a function (B) reflexive

View solution

Problem 25

Let \(R=\\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9)\) \((3,12),(3,6)\\}\) be a relation on the set \(A=\\{3,6,9,12\\}\) be a relation the set \(A=\\{3,6,9,12\\}\)

View solution

Key Concepts

Real Numbers

Real numbers form a set of numbers that include both rational numbers, like 2 or -5.5, and irrational numbers, such as \( \sqrt{2} \) and pi (\( \pi \)). They cover all points on the infinite number line, extending in both positive and negative directions. Real numbers can be classified into distinct categories:

• Natural Numbers: The counting numbers starting from 1, 2, 3, and so on.
• Whole Numbers: Natural numbers along with zero, so 0, 1, 2, 3, etc.
• Integers: All positive and negative whole numbers, including zero.
• Rational Numbers: Numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
• Irrational Numbers: Numbers that cannot be written as a simple fraction. These are often decimal numbers that do not terminate or repeat, like \( \sqrt{2} \) or \( \pi \).

Real numbers are essential for describing quantities that can have virtually any size or precision. They are vital in various mathematical fields, including calculus, algebra, and beyond.

Reflexivity

Reflexivity is one of the three critical properties an equivalence relation must satisfy. For a set to be reflexive, each element must relate to itself. Using the mathematical notation, for any element \( x \) in set \( R \), the pair \( (x, x) \) should exist in the relation. For example, considering set \( T = \{ (x, y): x-y \text{ is an integer} \} \), this property is satisfied because for any \( x \), \( x-x = 0 \) is indeed an integer. Thus, every real number is reflexive with itself in set \( T \). However, for set \( S = \{ (x, y): y = x + 1 \} \), reflexivity fails because \( x = x + 1 \) is never true for any real number \( x \). Therefore, this set does not satisfy the reflexivity condition. Reflexivity ensures that every item is in a self-loop when imagined as a vertex in a graph.

Symmetry

Symmetry requires that if an element \( x \) is related to \( y \), then \( y \) should also be related to \( x \). In simpler terms, if \( (x, y) \) is in the relation, \( (y, x) \) must also be part of it.In set \( T = \{ (x, y): x-y \text{ is an integer} \} \), this is satisfied. If \( x - y \) is an integer, then \( y - x \) is also an integer because it is just the negative of \( x - y \). Thus, symmetry holds true in \( T \).Contrarily, in set \( S = \{ (x, y): y = x + 1 \} \), symmetry fails. While \( y = x + 1 \) implies a particular direction from \( x \) to \( y \), reversing this statement results in \( x = y + 1 \), which doesn't hold true for the same pair, violating symmetry. Symmetry ensures a bidirectional relationship in pairs.

Transitivity

Transitivity is the property that if \( x \) is related to \( y \), and \( y \) is related to \( z \), then \( x \) should also be related to \( z \). It's like a chain reaction that connects elements across a set. In set \( T = \{ (x, y): x-y \text{ is an integer} \} \), transitivity holds. If \( x-y \) and \( y-z \) are both integers, then their sum \( x-z = (x-y) + (y-z) \) is also an integer, maintaining the transitive property.Conversely, in set \( S = \{ (x, y): y = x + 1 \} \), transitivity is absent. If \( y = x + 1 \) and \( z = y + 1 \), you get \( z = x + 2 \), not \( x + 1 \), which doesn’t fall back into set \( S \). This missing link shows that \( x \) and \( z \) aren't directly relatable through the same rule, thus, transitivity is not satisfied in set \( S \).