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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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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.

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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.
answered Feb 22, 2013 by meena.p
edited Mar 8, 2013 by balaji.thirumalai
 

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