Let A = {4,5,6} , B = {a,y,c,w}and R = {(4,c), (4,y), (5,a),(5,c), (5,w)} is a relation from A to B.Then we can write :

To analyze the relation $R$ from set $A$ to set $B$, we should recognize how the elements of $A$ map to the elements of $B$ through the pairs in $R$. The elements of $A$ are related to those in $B$ based on the given pairs. Let's break this down step-by-step.

1. ### Break Down the Problem

1. Identify the elements of sets $A$ and $B$:
   • $A = \{4, 5, 6\}$
   • $B = \{a, y, c, w\}$
2. Examine the relation $R$:
   • $R = \{(4,c), (4,y), (5,a), (5,c), (5,w)\}$
3. Determine which elements of $A$ relate to which elements of $B$.

2. ### Relevant Concepts

In set theory, a relation $R$ from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$, which consists of all possible pairs $(a, b)$ where $a \in A$ and $b \in B$.

3. ### Analysis and Detail

• The pairs in $R$ show how elements in $A$ correspond to those in $B$:
  • $4$ maps to $c$ and $y$ (i.e., $4$ is related to $c$ and $y$).
  • $5$ maps to $a$, $c$, and $w$ (i.e., $5$ is related to $a$, $c$, and $w$).
  • $6$ is not related to any element in $B$ (i.e., there are no pairs in $R$ with $6$).

4. ### Verify and Summarize

• Since there are no pairs including $6$, we can summarize the relation:
  • Element $4$ in $A$ relates to $c$ and $y$ in $B$.
  • Element $5$ in $A$ relates to $a$, $c$, and $w$ in $B$.
  • Element $6$ in $A$ does not relate to any element in $B$.

Final Answer

The relation $R$ can thus be expressed as:

• $4$ is related to $\{c, y\}$
• $5$ is related to $\{a, c, w\}$
• $6$ is related to $\emptyset$ (no relation).