Physical Education Class 12 Cbse Book Download Pdf ((FREE))
Physical Education Class 12 Cbse Book Download Pdf ((FREE))
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Physical Education Class 12 Cbse Book Download Pdf
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If $X$ is not a singleton, for $R in mathcal{R}$ let $R=R_x cup R_y$ where $x,y in X$. Prove $R$ is an equivalence relation if and only if $xRx$ and $yRy$
If $X$ is not a singleton, for $R in mathcal{R}$ let $R=R_x cup R_y$ where $x,y in X$. Prove $R$ is an equivalence relation if and only if $xRx$ and $yRy$.
My idea was to use the definitions of the equivalence relations.
$R$ is transitive:
If $xRy$, then $y in R_x$, and therefore there exists $z in X$ such that $y=xR_x=R_xR_z=R_xRz$, and so $xy=xR_xRz=xRz$, thus proving transitive.
$R$ is reflexive:
If $xRx$, then $R_x={x}$, thus $x=x$, thus $xRx$.
Now my idea was to use the definition of the least equivalence relation that contains $R$… but
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