Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b) : a, b \in A,\ b ext{ is exactly divisible by }a\}$. (i) Write $R$ in roster form (ii) Find the domain of $R$ (iii) Find the range of $R$.

Question

Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b) : a, b \in A,\ b	ext{ is exactly divisible by }a\}$. (i) Write $R$ in roster form (ii) Find the domain of $R$ (iii) Find the range of $R$.

Accepted Answer

> [!TLDR]
> The relation $R$ consists of all ordered pairs $(a,b)$ from $A$ where $a$ divides $b$ exactly. Writing it in roster form gives 10 pairs; the domain is $\{1,2,3,4,6\}$ and the range is $\{1,2,3,4,6\}$.

The key idea is simple: "b is exactly divisible by a" means a divides b with no remainder — in other words, b is a multiple of a. Since both numbers come from the same small set $A = \{1,2,3,4,6\}$, we can check every possible pair.

A Relation Arrow Diagram helps here. Imagine drawing two copies of set $A$, one on the left (for the first element $a$) and one on the right (for $b$). Draw an arrow from $a$ to $b$ whenever $a$ divides $b$. For example, 1 divides everything, so arrows go from 1 to every number. 2 divides 2, 4, and 6, so arrows go from 2 to those. 3 divides 3 and 6. 4 divides only 4. 6 divides only 6. This visual instantly shows you all the pairs — and also makes the domain and range obvious: the domain is the set of all left-side numbers that have at least one arrow leaving them, and the range is the set of all right-side numbers that have at least one arrow arriving.

Let’s now write it step by step.

1. **List all pairs $(a,b)$ where $a$ divides $b$.**  
   For each $a$ in $A$, find every $b$ in $A$ such that $b \div a$ is an integer (i.e., $b$ is a multiple of $a$).  
   - $a = 1$: every $b$ works, so $(1,1), (1,2), (1,3), (1,4), (1,6)$.  
   - $a = 2$: $b$ must be a multiple of 2 from $A$ → $2,4,6$ → $(2,2), (2,4), (2,6)$.  
   - $a = 3$: multiples of 3 in $A$ are $3,6$ → $(3,3), (3,6)$.  
   - $a = 4$: multiples of 4 in $A$ are just $4$ → $(4,4)$.  
   - $a = 6$: multiples of 6 in $A$ are just $6$ → $(6,6)$.

So $R = \{(1,1), (1,2), (1,3), (1,4), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (6,6)\}$.

> [!WARNING]
   > A common mistake is to forget that a number is always divisible by itself. Every $a$ gives at least $(a,a)$ — don’t leave those out.

2. **Find the domain of $R$.** …

Q.Let $A = \{1, 2, 3, 4, 6\}$ . Let $R$ be the relation on $A$ defined by $\{(a, b) : a, b \in A,\ b\text{ is exactly divisible by }a\}$ .

Concept understanding — Relation Arrow Diagram

Relation Arrow Diagram

The Intuition First

The Precise Definition

Key Points to Remember

A Relation on a Single Set

Q.Let A={1,2,3,4,6}A = \{1, 2, 3, 4, 6\}A={1,2,3,4,6}. Let RRR be the relation on AAA defined by {(a,b):a,b∈A, b is exactly divisible by a}\{(a, b) : a, b \in A,\ b\text{ is exactly divisible by }a\}{(a,b):a,b∈A, b is exactly divisible by a}.

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Q.Let $A = \{1, 2, 3, 4, 6\}$ . Let $R$ be the relation on $A$ defined by $\{(a, b) : a, b \in A,\ b\text{ is exactly divisible by }a\}$ .

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