Problem 21 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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 \(2: B=\\{(x, y) \in R \times R: x=\alpha y\) for some rational number \(\alpha\\}\) is an equivalence relation of \(R\). (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 is false

Step-by-Step Solution

Answer

Option C: Both statements are true, but Statement 2 doesn't explain Statement 1.

Step 1: Analyze Statement 1

To determine if set \(A\) is an equivalence relation, it must satisfy three properties: reflexivity, symmetry, and transitivity.\**Reflexivity**: For \( (x, x) \), \( y - x = x - x = 0 \), and 0 is an integer. Therefore, \( (x, x) \in A \) for all \( x \in R \). Reflexivity holds. **Symmetry**: If \( (x, y) \in A \), then \( y - x \) is an integer. So, \( x - y = -(y - x) \) is also an integer, meaning \( (y, x) \in A \). Symmetry holds. **Transitivity**: If \( (x, y) \in A \) and \( (y, z) \in A \), then \( y - x \) and \( z - y \) are integers. The sum \( (y - x) + (z - y) = z - x \) is also an integer, which implies \( (x, z) \in A \). Transitivity holds. \Since all three properties are satisfied, \(A\) is an equivalence relation. Thus, Statement 1 is true.

Step 2: Analyze Statement 2

Determine whether set \(B\) is an equivalence relation similar to set \(A\).\**Reflexivity**: For \( (x, x) \), \( x = 1 \cdot x \) where \( 1 \) is a rational number. Therefore, \( (x, x) \in B \) for all \( x \in R \). Reflexivity holds. **Symmetry**: If \( (x, y) \in B \), then \( x = \alpha y \) for some rational \( \alpha \). Consequently, \( y = \frac{1}{\alpha} x \) and since \( \alpha \) is rational, \( \frac{1}{\alpha} \) is also rational, meaning \( (y, x) \in B \). Symmetry holds. **Transitivity**: If \( (x, y) \in B \) and \( (y, z) \in B \), then \( x = \alpha y \) and \( y = \beta z \) for some rational \( \alpha, \beta \). So, \( x = (\alpha \beta) z \), where \( \alpha \beta \) is rational, implying \( (x, z) \in B \). Transitivity holds. \All three properties are satisfied, meaning \(B\) is an equivalence relation as well. Statement 2 is true.

Step 3: Evaluate the Explanation in Option B and Choose Best Option

Option B suggests Statement 2 explains Statement 1. However, the equivalence relation defined in Statement 2 does not serve as an explanation for the equivalence relation described in Statement 1. They both represent different criteria for being an equivalence relation. Thus, Option C is the better choice, stating both statements are true but independent explanations. Therefore, the solution to this problem is Option C: Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

Key Concepts

Real Numbers

Real numbers are a crucial concept in mathematics. They include all the numbers found on the number line. This means that they encompass integers, fractions (rational numbers), and even numbers that cannot be expressed as fractions, such as \(\sqrt{2}\), commonly known as irrational numbers.
Real numbers play an essential role in defining equivalence relations, as they provide a wide range of values to work with. By considering real numbers, we can delve into various mathematical constructs, including equivalence relations themselves.

• They include all rational numbers, such as \(\frac{1}{2}\) and \(-3\).
• They also include irrational numbers, like \(\pi\) and \(e\).
• Real numbers are used to define relations involving properties like distance and order.

Understanding real numbers is foundational for grasping advanced topics like equivalence relations, which often rely on these values to define certain mathematical properties.

Reflexivity

Reflexivity is a property required for a relation to be considered an equivalence relation. It's quite simple: every element must relate to itself. For a set of real numbers, this means that given any real number \(x\), the pair \( (x, x) \) must belong to the relation.
Reflexivity illustrates the idea that elements are inherently related to themselves in a given context. For example:

• In the set \(A = \{(x, y) \in R \times R: y-x \text{ is an integer}\}\), any real number \(x\) will always satisfy \(y-x=0\).
• The difference \(y-x\) results in \(0\), which is definitely an integer.

Without reflexivity, a relation cannot be termed an equivalence relation, as it's one of the core components that assure internal consistency within the set.

Symmetry

The symmetry property in a relation refers to the idea that if an element \(x\) is related to an element \(y\), then \(y\) must also be related to \(x\). This property ensures the equality and mutual respect in the relationship between elements.
In mathematical terms, for any two elements \(x\) and \(y\) within an equivalence relation, if \( (x, y) \) belongs to the relation, then so should \( (y, x) \).

• For the set \(A = \{(x, y) \in R \times R: y-x \text{ is an integer}\}\), if \( x - y = -(y - x)\), it's a negative integer, which is still an integer.
• In another example, in set \(B = \{(x, y) \in R \times R: x=\alpha y, \alpha \text{ is rational}\}\), knowing \(x = \alpha y\) implies \(y = \frac{1}{\alpha} x\), which remains rational.

Ensuring symmetry is vital for a full-fledged equivalence relation, affirming its balanced nature.

Transitivity

Transitivity is the property of a relation that ensures if one element relates to a second, and the second relates to a third, then the first must relate to the third. This is critical for the cohesive structure in equivalence relations.
For example, if you have elements \(x, y,\) and \(z\):

• In set \(A=\{(x, y) \in R \times R: y-x \text{ is an integer}\}\), if both \( y-x \) and \( z-y \) are integers, then \( z-x \) must also be an integer.
• In set \(B=\{(x, y) \in R \times R: x=\alpha y, \alpha \text{ is rational}\}\), for \(x = \alpha y\) and \(y = \beta z\), the product \(\alpha \beta \) remains rational, showing that \(x\) relates directly to \(z\).

This cascading effect of relations ensures that equivalence relations are complete and consistent across the elements within the set they define. Transitivity, along with reflexivity and symmetry, consolidates the foundation of equivalence relations.