To determine the relation R, we need to find all ordered pairs $(a, b)$ such that $a$ and $b$ are elements of the set A = $\{1, 2, 3, 4\}$, and $a$ divides $b$. The condition "a divides b" means that when $b$ is divided by $a$, the remainder is zero. We will systematically check each element of A as $a$ and then as $b$ to form the ordered pairs.

Relation Definition

The given set is A = $\{1, 2, 3, 4\}$.

The relation R is defined as R = $\{(a, b); a \text{ divides } b\}$, where $a \in \text{A}$ and $b \in \text{A}$.

Let's find all the ordered pairs $(a, b)$ that satisfy this condition.

Finding Ordered Pairs in Relation R

We will go through each element $a$ in set A and find all elements $b$ in set A for which $a$ divides $b$.

• When $a = 1$:
  • $1$ divides $1$ (since $1 \div 1 = 1$, remainder $0$). So, $(1, 1)$ is in R.
  • $1$ divides $2$ (since $2 \div 1 = 2$, remainder $0$). So, $(1, 2)$ is in R.
  • $1$ divides $3$ (since $3 \div 1 = 3$, remainder $0$). So, $(1, 3)$ is in R.
  • $1$ divides $4$ (since $4 \div 1 = 4$, remainder $0$). So, $(1, 4)$ is in R.

  Pairs from $a=1$: $(1, 1), (1, 2), (1, 3), (1, 4)$.

• When $a = 2$:
  • $2$ divides $1$? No ($1 \div 2$ gives remainder $1$).
  • $2$ divides $2$ (since $2 \div 2 = 1$, remainder $0$). So, $(2, 2)$ is in R.
  • $2$ divides $3$? No ($3 \div 2$ gives remainder $1$).
  • $2$ divides $4$ (since $4 \div 2 = 2$, remainder $0$). So, $(2, 4)$ is in R.

  Pairs from $a=2$: $(2, 2), (2, 4)$.

• When $a = 3$:
  • $3$ divides $1$? No.
  • $3$ divides $2$? No.
  • $3$ divides $3$ (since $3 \div 3 = 1$, remainder $0$). So, $(3, 3)$ is in R.
  • $3$ divides $4$? No ($4 \div 3$ gives remainder $1$).

  Pairs from $a=3$: $(3, 3)$.

• When $a = 4$:
  • $4$ divides $1$? No.
  • $4$ divides $2$? No.
  • $4$ divides $3$? No.
  • $4$ divides $4$ (since $4 \div 4 = 1$, remainder $0$). So, $(4, 4)$ is in R.

  Pairs from $a=4$: $(4, 4)$.

Constructing the Relation R

Combining all the ordered pairs found above, the relation R is:

Comparing with Options

Let's compare this derived set R with the given options:

Based on the comparison, Option 3 is the correct set of ordered pairs for the relation R where $a$ divides $b$.