Question

If A = {2, 4, 6}, then domain of the relation R = {(a, b): a, b ∈ A, |a| - |b| is even number} defined on A is

• A. {2,4}
• B. {4,6}
• C. {2,6}
• D. {2,4,6}

Answer 1

Answer: (D)

{2,4,6}

Answer 2

The relation is defined as $R = \{(a, b): a, b \in A, |a| - |b| \text{ is an even number}\}$. The set is $A = \{2, 4, 6\}$.

The condition for an ordered pair $(a, b)$ to be in $R$ is that $|a| - |b|$ must be an even number. This implies that $|a|$ and $|b|$ must have the same parity (both even or both odd).

For any integer $x$, $|x|$ has the same parity as $x$. Therefore, the condition "$|a| - |b|$ is even" is equivalent to "$a$ and $b$ have the same parity".

Let's examine the elements of set $A = \{2, 4, 6\}$:

• 2 is an even number.
• 4 is an even number.
• 6 is an even number.

All elements in set $A$ are even. So, for any $a \in A$ and any $b \in A$, both $a$ and $b$ are even. This means they always have the same parity. Consequently, the condition "$a$ and $b$ have the same parity" is always satisfied for any $a, b \in A$.

This means that every possible ordered pair $(a, b)$ where $a \in A$ and $b \in A$ will satisfy the condition and belong to the relation $R$. Thus, the relation $R$ is the Cartesian product of $A$ with itself, i.e., $R = A \times A$. $R = \{(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)\}$.

The domain of a relation is the set of all first elements of the ordered pairs in the relation. For $R = A \times A$, the first element $a$ can be any element from set $A$. Therefore, the domain of $R$ is $A$.

Domain$(R) = \{2, 4, 6\}$.