If R and S are symmetric relations on the set A, then prove that R intersection S and R union S are symmetric.

Asked by abhinavsaini123 | 1st Jun, 2015, 08:57: PM

Expert Answer:

I t space i s space g i v e n space t h a t space R space a n d space S space a r e space s y m m e t r i c. l e t space u s space t a k e space left parenthesis x comma y right parenthesis space element of open parentheses R intersection S close parentheses rightwards double arrow left parenthesis x comma y right parenthesis element of R space a n d space left parenthesis x comma y right parenthesis space element of S rightwards double arrow left parenthesis y comma x right parenthesis space element of R space a n d space left parenthesis y comma x right parenthesis space element of S. S i n c e space R space a n d space S space a r e space s y m m e t r i c. therefore R intersection S space i s space s y m m e t r i c.  l e t space u s space t a k e space left parenthesis a comma b right parenthesis space element of open parentheses R union S close parentheses space rightwards double arrow space left parenthesis b comma a right parenthesis space element of e i t h e r space R space o r space S. space S i n c e space R thin space a n d space S space a r e space s y m m e t r i c. I t space a l s o space i m p l i e s space left parenthesis b comma a right parenthesis space element of open parentheses R union S close parentheses space therefore R union S space i s space s y m m e t r i c.

Answered by Prasenjit Paul | 2nd Jun, 2015, 10:50: AM

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