The problem you uploaded asks for the relation $R \circ R^{-1}$ based on a given set and relation $R$.

Problem:

• Set A = {1, 2, 3, 4} and Set B = {1, 3, 5}.
• Relation $R$ is defined such that $(a, b) \in R \iff a < b$ (meaning $a$ is less than $b$).

You are asked to compute $R \circ R^{-1}$, where:

• $R$ is the relation of elements of $A$ being less than elements of $B$,
• $R^{-1}$ is the inverse relation of $R$, i.e., $(b, a) \in R^{-1} \iff (a, b) \in R$,
• $R \circ R^{-1}$ is the composition of $R$ and $R^{-1}$, which means we are looking for all pairs $(a, b)$ such that $aRb$ and $bR^{-1}c$, resulting in $aRc$.

Steps to find $R \circ R^{-1}$:

1. Find the pairs in relation $R$: From $R$ as $a < b$, we check all pairs:

   • $R = \{(1, 3), (1, 5), (2, 3), (2, 5), (3, 5)\}$.

2. Find the inverse $R^{-1}$: In $R^{-1}$, the pairs are swapped from $R$:

   • $R^{-1} = \{(3, 1), (5, 1), (3, 2), (5, 2), (5, 3)\}$.

3. Compute $R \circ R^{-1}$: This involves combining $R$ and $R^{-1}$. Specifically, we look for all possible $(a, c)$ such that there is a $b$ where $(a, b) \in R$ and $(b, c) \in R^{-1}$.

   The relevant pairs will be formed by chaining the relations:

   • From $(1, 3) \in R$ and $(3, 1) \in R^{-1}$, we get $(1, 1)$.
   • From $(1, 5) \in R$ and $(5, 1) \in R^{-1}$, we get $(1, 1)$ again (already included).
   • From $(2, 3) \in R$ and $(3, 1) \in R^{-1}$, we get $(2, 1)$.
   • From $(2, 5) \in R$ and $(5, 1) \in R^{-1}$, we get $(2, 1)$ (already included).
   • From $(3, 5) \in R$ and $(5, 3) \in R^{-1}$, we get $(3, 3)$.

   So, the result for $R \circ R^{-1}$ is:

   • $\{(1, 1), (2, 1), (3, 3)\}$.

Answer:

• Based on the options given, none of the options exactly match. But if we look closely at the options, $(c)$ has pairs $(3, 3), (3, 5), (5, 3), (5, 5)$, which is the closest fit given the pattern we see. Thus, the relation $R \circ R^{-1}$ most closely resembles option (c).

Would you like further details on this process or another clarification?

5 Related Questions:

1. What is the difference between $R$ and $R^{-1}$ in relational mathematics?
2. How do we interpret relational compositions like $R \circ R^{-1}$ in set theory?
3. Why are only certain pairs included in the composition of two relations?
4. What is the significance of finding inverse relations in set theory?
5. How do properties like transitivity play a role in relations and their inverses?

Tip:

When working with relations and compositions, drawing a diagram or table of the pairs can help visualize the process, especially when dealing with more complex compositions like $R \circ R^{-1}$.