# Let $$R$$ be the relation in the set $$\{1, 2, 3, 4\}$$ given by $$R = \{(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)\}$$. Choose the correct answer.

(A) $$R$$ is reflexive and symmetric but not transitive. (B) $$R$$ is reflexive and transitive but not symmetric. (C) $$R$$ is symmetric and transitive but not reflexive. (D) $$R$$ is an equivalence relation.

Toolbox:
• A relation R in a set A is a equivalance relation if it is symmetric, reflexive and transitive.
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
Given $R=\{(1,2),(2,2),(1,1),(4,4),(1,3),(3,3)(3,2)\}$ is the relation in the set $\{1,2,3,4\}$:
Given the set $\{1,2,3,4\}$, For every element of the set, $(1,1), (2,2), (3,3), (4,4) \in R$. Hence $R$ is reflexive.
Given the set $\{1,2,3,4\}$, we can observe that while $(1,2) \in R$, $(2,1) \not \in R$. Hence $R$ is not symmetric.
Given the set $\{1,2,3,4\}$, we can observe by looking at the ordered pairs that if $(a,b) \in R$ and $(b,c) \in R \rightarrow (a,c) \in R$.
Therefore, $R$ is transitive.
Therefore, $R$ is transitive and reflexive but not symmetric.
edited Mar 8, 2013