Find the transitive closure of the relation)}1,4(),4,3(),3,2(),2,1{(R . ShowiR for allvalues ofi that give new elements of the transitive closure

1. Break Down the Problem

We need to find the transitive closure of the relation $R = \{(1,4), (4,3), (3,2), (2,1)\}$. The transitive closure of a relation is the smallest relation that contains $R$ and is transitive. We will examine pairs to see if new connections can be formed.

2. Relevant Concepts

The transitive property states that if $(a,b) \in R$ and $(b,c) \in R$, then $(a,c)$ should also be in the transitive closure. We will apply this property iteratively.

3. Analysis and Detail

Starting with the relation:

• Initial pairs in $R$:
  • $(1, 4)$
  • $(4, 3)$
  • $(3, 2)$
  • $(2, 1)$

Next, we look for new pairs using the transitive property:

1. From $(1, 4)$ and $(4, 3)$:

   • We can conclude $(1, 3)$.

2. From $(1, 3)$ and $(3, 2)$:

   • We can conclude $(1, 2)$.

3. From $(1, 2)$ and $(2, 1)$:

   • We can conclude $(1, 1)$.

4. From $(4, 3)$ and $(3, 2)$:

   • We can conclude $(4, 2)$.

5. From $(4, 2)$ and $(2, 1)$:

   • We can conclude $(4, 1)$.

6. From $(3, 2)$ and $(2, 1)$:

   • We can conclude $(3, 1)$.

7. From $(2, 1)$ and $(1, 4)$:

   • We can conclude $(2, 4)$.

After iterating through these pairs, we compile a complete list of pairs in the transitive closure $R^*$:

4. Verify and Summarize

The resulting transitive closure $R^*$ is: $R^* = \{(1, 4), (4, 3), (3, 2), (2, 1), (1, 3), (1, 2), (1, 1), (4, 2), (4, 1), (3, 1), (2, 4)\}$

Final Answer

The new elements added to the transitive closure are:

• $(1, 3)$
• $(1, 2)$
• $(1, 1)$
• $(4, 2)$
• $(4, 1)$
• $(3, 1)$
• $(2, 4)$

So the transitive closure $R^*$ is: $R^* = \{(1, 4), (4, 3), (3, 2), (2, 1), (1, 3), (1, 2), (1, 1), (4, 2), (4, 1), (3, 1), (2, 4)\}$