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# Each of the following defines a relation on N : \begin{array}{1 1} (i)\quad x\; is\; greater\; than\; y,x,y\quad N\\(ii)\quad x+y=10,x,y\quad N\\(iii)\quad x\;y\;is\;square\; of\; an\; integer\;x,y\quad N\\(iv)\quad x+4y=10\;x,y\quad N\end{array}Determine which of the above relations are reflexive,symmetric and transitive.

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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 R defined by

$R \{(x,y):x \;is\; greater\; than\; y\; \qquad x,y \in N\}$

Consider (1,1) one cannot be greater than for every element $x \in N$

$x > x$

Hence R is not reflexive

Consider $(3,2) \in R$ ie 3 is greater than 2

but $(2,3) \notin$ as 2 is not greater than 3

R is not symmetric

Consider $(3,2),(2,1) \in R$

ie $3 > 2 \;and \;2 > 1$

$=>3 > 1$

Hence $(3,1) \in R$

R is transitive

Solution:R is transitive but not reflexive and not symmetric

answered Mar 4, 2013 by
edited Mar 27, 2013 by meena.p