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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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If A={1,2,3,4},define relations on A which have properties of being:\begin{array}{1 1}(a)\quad reflexive,transitive\;but\;not\;symmetric & \;\\(b)\quad symmetric\; but \;neither\;reflexive\;nor\;transitive & \;\\(c)\quad reflexive,symmetric\;and\;transitive\end{array}

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Toolbox:
  • 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 $A=\{1,2,3,4\}$
Let $R_1=\{(1,1),(2,2),(3,3),(4,4),(2,3),(3,2)(4,2),(4,3)\}$
Step1: $R_1$ is reflexive,transitive but not symmetric
Since $(a,a) \in R\; for\; all\;a \in \{1,2,3,4\}$
$R_1$ is reflexive
Since $(4,2) \in R \;but\; (2,4) \notin R$
$R_1$ is not symmetric
Since $(4,3) \in R , (3,2) \in R => (4,2) \in R $
$R_1$ is transitive
$R_1$ is reflexive,transitive but not symmetric
Step2: $R_2$ is symmetric but neither reflexive nor transitive
Let $R_2=\{(1,1),(2,3),(3,2),(1,2)\}$
Since $(2,2),(3,3)(4,4) \notin R$
$R_2$ is not reflexive
Since $(2,3) \in R => (3,2) \in R $
$R_2$ is symmetric
Since $(1,2) \in R and (2,3) \in R but (1,3) \notin R $
$R_2$ is not transitive
$R_2 is symmetric but neither reflxive nor transitive
Step3: $R_3$ is reflexive,symmetric and transitive
Let $R_3=\{(1,1),(2,2),(3,3),(4,4),(2,3),(3,2)(2,4),(4,3),(3,4)(4,2)\}$
Since $(1,1),(2,2)(3,3)(4,4)\;all \in R_3$
$R_3$ is reflexive
Since $(2,3),(3,2)(2,4)(4,2)(4,3)(3,4) \in R$
$R_3$ is symmetric
Since $(2,3) \in R (3,4) \in R => (2,4) \in R $
$R_3$ is transitive
$R_3$ is reflexive,symmetric and transitive
answered Mar 3, 2013 by meena.p
edited Mar 26, 2013 by meena.p
 

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