Q5(ii):

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

Step 1: Formal Definition of the Relation

Given the set $A = \{1, 2, 3, 4, 6\}$, the relation $R$ is defined as:

$R = \{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}$

The condition "$b$ is exactly divisible by $a$" implies that $\frac{b}{a} \in \mathbb{Z}$, or equivalently, $a$ is a divisor of $b$. [By the definition of divisibility in number theory].

Step 2: Systematic Identification of Ordered Pairs

We test each element $a \in A$ to determine which elements $b \in A$ satisfy the condition $a|b$:

• For $a = 1$: $1$ divides $1, 2, 3, 4, 6$. Pairs: $(1, 1), (1, 2), (1, 3), (1, 4), (1, 6)$.
• For $a = 2$: $2$ divides $2, 4, 6$. Pairs: $(2, 2), (2, 4), (2, 6)$.
• For $a = 3$: $3$ divides $3, 6$. Pairs: $(3, 3), (3, 6)$.
• For $a = 4$: $4$ divides $4$. Pair: $(4, 4)$.
• For $a = 6$: $6$ divides $6$. Pair: $(6, 6)$.

The complete set $R$ is:

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

Step 3: Determination of the Domain

The domain of a relation $R$ is defined as the set of all first components of the ordered pairs in $R$. [Per the definition of the Domain of a Relation: $Dom(R) = \{a \in A : \exists b \in A \text{ such that } (a, b) \in R\}$].

From the set $R$ derived in Step 2, the unique first components are $\{1, 2, 3, 4, 6\}$.

Step 4: Verification via Mapping Visualization

Since every element $a \in A$ appears as a first component in at least one ordered pair (specifically, $(a, a) \in R$ for all $a \in A$ due to the reflexive property of the "divides" relation), the domain is the entire set $A$.

Final Solution: The domain of $R$ is $\{1, 2, 3, 4, 6\}$.