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Show that the relation $R$ in the set $A=\{x\in Z: 0 \leq x \leq 12\}$, given by $(ii) R = \{(a,b): a=b\}$ is an equivalence relation. Find the set of all elements related to 1.

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  • A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transitive.
  • A relation R in a set A is called reflexive. if $(a,a) \in R\;for\; all\; a\in A$
  • A relation R in a set A is called 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 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 set $A=\{x \in Z:0 \leq x\leq 12\}$ and relation $R=\{(a,b):a=b)\}$:
Let $a=b$. For every $a \in A (a,a) \in R$, $a=a$. Therefore R is reflexive.
If $(a,b) \in R$ then $a=b$. Similarly, if $(b,a) \in R$, then $b=a$. Therefore, R is symmetric.
If $(a,b) \in R$ then $a=b$ and, if $(b,c) \in R$, then $b=c$.
Now, if $(a,c) \in R$ then $a=c$. Since $a=b$ and $b=c$ $\rightarrow$ $a=c$. Therefore, R is transitive..
Hence $R$ is an equivalence relation as its reflexive, symmetric and transitive.
Given a set $A=\{x \in Z:0 \leq x\leq 12\}$, the set of elements in $A$ is $\{0,1,2,3,4,5,6,7,8,9,10,11,12\}$
Among these elements the ones that are related to 1 are those that satisfy the condition $a=1$.
We can observe that only element that satisfies this condition is $\{1\}$.
answered Mar 8, 2013 by balaji.thirumalai

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