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Standard XII

Mathematics

Transitive Relations

For any relat...

Question

For any relation R in any set A, we can define the inverse relation R^-1 by a relation a R^-1 b if and only if bRa.Prove that R is symmetric if and only if R=R^-1

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Solution

Let us assume that R is symmetric. Then if xRy then yRx.
(x,y) $\in$ R
if and only if xRy
if and only if yRx (because R is symmetric)
if and only if (y,x) $\in$ R
if and only if x R^-1 y
Hence R= R^-1

Conversely suppose that R= R^-1
Let xRy
(x,y) $\in$ R

(y,x) $\in$ R^-1
(y,x) $\in$ R(because R= R^-1)
So, R is symmetric.

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For any relation R in any set A, we can define the inverse relation R-1 by a relation a R-1 b if and only if bRa.Prove that R is symmetric if and only if R=R-1

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For any relation R in any set A, we can define the inverse relation R^-1 by a relation a R^-1 b if and only if bRa.Prove that R is symmetric if and only if R=R^-1