To find $R \circ R$ (also known as the composition of the relation $R$ with itself), follow these steps:

What is $R \circ R$?

The composition $R \circ R$ of a relation $R$ on a set $A$ means applying the relation $R$ twice. Formally, $R \circ R$ is a relation on $A$ such that: $(x, z) \in R \circ R \iff \exists y \in A \text{ such that } (x, y) \in R \text{ and } (y, z) \in R.$ In simple terms, for each pair $(x, z)$ in $R \circ R$, there must be some element $y \in A$ where:

1. $x$ is related to $y$, and
2. $y$ is related to $z$.

Steps to Find $R \circ R$:

1. List the pairs in $R$: Identify all ordered pairs $(a, b)$ in $R$.
2. Check the conditions for $R \circ R$: For each pair $(x, y)$ in $R$, find if there exists another pair $(y, z)$ in $R$.
3. Form the new pairs: For each valid pair, create a new pair $(x, z)$ that satisfies both conditions.
4. Collect all new pairs: The set of all such pairs $(x, z)$ will be $R \circ R$.

Example:

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

To find $R \circ R$:

• Check pairs where the second element of one pair matches the first element of another.
  • From $(1, 2)$ and $(2, 3)$, we get $(1, 3)$.
  • From $(1, 2)$ and $(2, 4)$, no match.
  • From $(2, 3)$ and $(3, 4)$, we get $(2, 4)$.

Thus, $R \circ R = \{(1, 3), (2, 4)\}$.

Would you like more details on this process or a different example?

Related Questions:

1. How do you find the inverse of a relation?
2. What is the difference between the composition of relations and functions?
3. Can you explain the properties of relation composition?
4. How does the identity relation interact with composition?
5. What are some common uses of relation composition in mathematics?

Tip:

When dealing with compositions of relations, drawing a diagram or mapping the elements can make it easier to visualize how the elements are connected.