Analyzing Relation Properties on Natural Numbers

The question asks us to consider a relation R defined on the set of natural numbers $\mathbb{N}$ by the rule $xRy$ if $x^2 - 5xy + 4y^2 = 0$. We need to determine which of the given statements about this relation R (reflexive, symmetric, transitive) are correct.

Understanding the Relation R

The defining equation for the relation R is $x^2 - 5xy + 4y^2 = 0$. Let's factor this equation to understand the condition for $xRy$ more clearly:

$\qquad x^2 - 5xy + 4y^2 = 0$
$\qquad x^2 - xy - 4xy + 4y^2 = 0$
$\qquad x(x - y) - 4y(x - y) = 0$
$\qquad (x - y)(x - 4y) = 0$

This equation is satisfied if and only if $x - y = 0$ or $x - 4y = 0$. Therefore, the relation R is defined as:

$\qquad xRy \iff x = y \text{ or } x = 4y$

This means that for any two natural numbers $x$ and $y$, $x$ is related to $y$ if $x$ is equal to $y$ or if $x$ is four times $y$. The set of natural numbers is $\mathbb{N} = \{1, 2, 3, 4, \dots\}$.

Checking for Reflexivity

A relation R on a set A is reflexive if $aRa$ for every element $a \in A$. In this case, we need to check if $xRx$ for every natural number $x \in \mathbb{N}$.

According to our relation definition, $xRx$ means $(x = x \text{ or } x = 4x)$.

• The condition $x = x$ is always true for any number $x$.
• The condition $x = 4x$ implies $3x = 0$, which means $x = 0$. However, $x$ must be a natural number ($x \in \mathbb{N}$), and $0$ is not a natural number. So, $x=4x$ is never true for $x \in \mathbb{N}$.

Since the condition "$x = x$" is always true, the condition "$x = x$ or $x = 4x$" is always true for any $x \in \mathbb{N}$.

Thus, $xRx$ holds for all $x \in \mathbb{N}$. The relation R is reflexive.

Statement 1: R is reflexive - Correct.

Checking for Symmetry

A relation R on a set A is symmetric if whenever $aRb$ holds, $bRa$ also holds, for all $a, b \in A$. We need to check if $xRy \implies yRx$ for all $x, y \in \mathbb{N}$.

Suppose $xRy$ holds. This means $(x = y \text{ or } x = 4y)$. We need to see if this implies $(y = x \text{ or } y = 4x)$.

• Case 1: If $x = y$. If $x=y$, then $y=x$. The condition $(y = x \text{ or } y = 4x)$ becomes $(y = y \text{ or } y = 4y)$, which is true because $y=y$ is true. So, symmetry holds when $x=y$.
• Case 2: If $x = 4y$. We need to check if this implies $(y = x \text{ or } y = 4x)$.
  • Is $y = x$? Substituting $x = 4y$, we get $y = 4y$, which means $3y = 0$, or $y = 0$. Since $y \in \mathbb{N}$, $y$ cannot be $0$. So $y=x$ is false if $x=4y$ (for $y \in \mathbb{N}$).
  • Is $y = 4x$? Substituting $x = 4y$, we get $y = 4(4y)$, which means $y = 16y$, or $15y = 0$, or $y = 0$. Since $y \in \mathbb{N}$, $y$ cannot be $0$. So $y=4x$ is false if $x=4y$ (for $y \in \mathbb{N}$).

  Since neither $y=x$ nor $y=4x$ is true when $x=4y$ (for $y \in \mathbb{N}$), $yRx$ does not hold when $x = 4y$.

Let's take a counterexample from $\mathbb{N}$. Let $y=1$. Then $x=4y=4$. So $4R1$ holds because $4 = 4 \times 1$.

Now let's check if $1R4$ holds. $1R4 \iff (1 = 4 \text{ or } 1 = 4 \times 4)$. Both conditions ($1=4$ and $1=16$) are false.

Since $4R1$ holds but $1R4$ does not hold, the relation R is not symmetric.

Statement 2: R is symmetric - Incorrect.

Checking for Transitivity

A relation R on a set A is transitive if whenever $aRb$ and $bRc$ hold, $aRc$ also holds, for all $a, b, c \in A$. We need to check if $xRy$ and $yRz \implies xRz$ for all $x, y, z \in \mathbb{N}$.

Suppose $xRy$ and $yRz$ hold. $xRy \iff (x = y \text{ or } x = 4y)$ $yRz \iff (y = z \text{ or } y = 4z)$

We need to check if this implies $(x = z \text{ or } x = 4z)$. Let's examine the possible combinations:

• Case 1: $x = y$ and $y = z$. This implies $x = z$. Since $x=z$, $xRz$ holds.
• Case 2: $x = y$ and $y = 4z$. This implies $x = 4z$. Since $x=4z$, $xRz$ holds.
• Case 3: $x = 4y$ and $y = z$. This implies $x = 4z$. Since $x=4z$, $xRz$ holds.
• Case 4: $x = 4y$ and $y = 4z$. Substituting the second equation into the first gives $x = 4(4z) = 16z$. Now we need to check if $xRz$ holds when $x = 16z$. $xRz \iff (x = z \text{ or } x = 4z)$.
  • Is $x = z$? $16z = z \implies 15z = 0 \implies z = 0$. Not possible for $z \in \mathbb{N}$.
  • Is $x = 4z$? $16z = 4z \implies 12z = 0 \implies z = 0$. Not possible for $z \in \mathbb{N}$.

  Since neither $x=z$ nor $x=4z$ holds when $x=16z$ (for $z \in \mathbb{N}$), $xRz$ does not hold in this case.

Let's take a counterexample from $\mathbb{N}$ for Case 4. Let $z=1$. Then $y=4z=4$. Then $x=4y=4(4)=16$.

• $xRy$: $16R4$. This holds because $16 = 4 \times 4$.
• $yRz$: $4R1$. This holds because $4 = 4 \times 1$.
• $xRz$: $16R1$. This means $(16 = 1 \text{ or } 16 = 4 \times 1)$. Both conditions ($16=1$ and $16=4$) are false. So $16R1$ does not hold.

Since $16R4$ and $4R1$ hold, but $16R1$ does not hold, the relation R is not transitive.

Statement 3: R is transitive - Incorrect.

Conclusion

Based on our analysis:

• Statement 1 (R is reflexive) is Correct.
• Statement 2 (R is symmetric) is Incorrect.
• Statement 3 (R is transitive) is Incorrect.

Therefore, only statement 1 is correct.

Revision Table: Properties of Relation R

Here is a summary of the findings regarding the properties of the relation R on $\mathbb{N}$ defined by $xRy \iff x^2 - 5xy + 4y^2 = 0 \iff x=y$ or $x=4y$.

Additional Information: Types of Relations

Relations on a set A are fundamental concepts in mathematics. They describe how elements within the set are connected. We examined three key properties:

• Reflexive Relation: Every element is related to itself. Think of the equals relation ($=$) on any set, or "is less than or equal to" ($\le$) on numbers.
• Symmetric Relation: If element A is related to element B, then element B must also be related to element A. Examples include "is a sibling of" (assuming sibling is a mutual relationship) or "is equal to" ($=$). "Is less than" ($<$) is not symmetric because $1 < 2$ but $2 < 1$ is false.
• Transitive Relation: If element A is related to element B, and element B is related to element C, then element A must also be related to element C. Examples include "is less than" ($<$), "is equal to" ($=$), or "is an ancestor of" (if A is an ancestor of B, and B is an ancestor of C, then A is an ancestor of C).

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. The relation R discussed here is reflexive but neither symmetric nor transitive, so it is not an equivalence relation.