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# True or False: Every relation which is symmetric and transitive is also reflexive.

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Toolbox:
• 1.For a given function R in A.
• 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$\; a_1,a_2,a_3 \in A$
• 2. we show by example the given statment is to not true
Let $A=\{-5,-6\}$

Define $R=\{(-5,-6)(-6,-5),(-5,-5)\}$

R is not reflexive since $(-6,-6) \in R$

R is symmetric since $(-5,-6) \in R=>(-6,-5) \in R$

R is transitive since $(-5,-6),(-6,-5) \in R=>(-5,-5) \in R$

Hence R is a relation it is symmetric,transitive but not reflexive

Symmetric & transiive dies not imply reflexive

$'False'$

answered Mar 6, 2013 by
edited Mar 28, 2013 by meena.p

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