Q.Let the relation $R$ be defined in $\mathbb{N}$ by $aRb$ if $2a + 3b = 30$. Then $R =$ ______.

Concept: Relation Properties — We find all ordered pairs $(a,b) \in \mathbb{N}^2$ that satisfy the given linear equation.

Step 1: The condition is $2a + 3b = 30$, with $a, b \in \mathbb{N}$ (natural numbers, usually starting from 1). Solve for $a$:

$a = \frac{30 - 3b}{2}$.

Step 2: For $a$ to be a natural number, $30 - 3b$ must be positive and even.

$30 - 3b > 0 \Rightarrow b < 10$, so $b \in \{1,2,\dots,9\}$.

Also $30 - 3b$ even $\Rightarrow$ $3b$ even $\Rightarrow$ $b$ even (since 3 is odd). So $b \in \{2,4,6,8\}$. …

We are working with the set of natural numbers $\mathbb{N}$. Usually in Indian exams, $\mathbb{N} = \{1, 2, 3, \dots\}$ (positive integers). The relation $R$ is defined by the condition $2a + 3b = 30$, where $a$ and $b$ both belong to $\mathbb{N}$.

The key idea: we need to find all pairs $(a,b)$ of natural numbers that satisfy this linear Diophantine equation. Since the equation is linear and the coefficients are small, we can solve by expressing one variable in terms of the other and then checking which values keep both variables natural.

1. Express $a$ in terms of $b$ From $2a + 3b = 30$, we get

For $a$ to be a natural number, $30 - 3b$ must be positive and even, and $a$ must be at least $1$.

2. Determine the range of $b$ Since $a \geq 1$, we have

Also $b \geq 1$ (natural number). So $b$ can be $1, 2, 3, \dots, 9$.

3. Check which $b$ give integer $a$

   $a$ is integer exactly when $30 - 3b$ is even. Since $30$ is even, $3b$ must be even. $3b$ is even only when $b$ is even (because $3$ is odd). So $b$ must be even.

   From $1$ to $9$, the even numbers are $2, 4, 6, 8$.

4. Compute corresponding $a$ for each even $b$

   • If $b = 2$: $a = \frac{30 - 6}{2} = \frac{24}{2} = 12$ → $(12,2)$
   • If $b = 4$: $a = \frac{30 - 12}{2} = \frac{18}{2} = 9$ → $(9,4)$
   • If $b = 6$: $a = \frac{30 - 18}{2} = \frac{12}{2} = 6$ → $(6,6)$ …

Method: Writing a Relation in Roster Form From an Equation

Use this when a relation on $\mathbb{N}$ (or $\mathbb{Z}$) is given by an equation like $2a+3b=k$ and you must list all its pairs.

Steps

Step 1: Solve for one variable in terms of the other.

For example, $a=\dfrac{k-3b}{2}$, so $a$ depends on $b$.

Step 2: Impose the domain conditions. …

Common Mistakes

Mistake 1: Including a pair with a coordinate of $0$.

Why it's wrong: in the usual Indian-syllabus convention $\mathbb{N}=\{1,2,3,\dots\}$, so the solution $b=10,\ a=0$ is not allowed. Correct approach: discard pairs where either coordinate is $0$.

Mistake 2: Missing the divisibility or parity restriction. …