L e t space R space b e space a space r e l a t i o n space i n space t h e space s e t space o f space r e a l space n u m b e r s space a n d space R equals open curly brackets left parenthesis x comma y right parenthesis colon space x less or equal than y cubed close curly brackets. space G i v e space a n space e x a m p l e space t h a t space f o r space a n y space x comma y comma z element of R comma space left parenthesis x comma y right parenthesis element of R comma space left parenthesis y comma z right parenthesis element of R space b u t space left parenthesis y comma z right parenthesis space d o e s space n o t space b e l o n g space t o space R.

Asked by surajsubedi14 | 3rd Mar, 2015, 10:09: AM

Expert Answer:

C o n s i d e r space t h e space n u m b e r s space 4 space a n d space 2. S i n c e comma space 4 less or equal than 2 cubed comma space w e space h a v e comma space open parentheses 4 comma 2 close parentheses element of R. A l s o comma space w e space h a v e comma N o w space c o n s i e r space t h e space n u m b e r s comma space 2 space a n d space 2 to the power of 1 third end exponent S i n c e comma space 2 less or equal than open parentheses 2 to the power of 1 third end exponent close parentheses cubed comma space w e space h a v e comma space open parentheses 2 comma 2 to the power of 1 third end exponent close parentheses element of R N o w space c o n s i d e r space 4 space a n d space 2 to the power of 1 third end exponent. S i n c e space 4 greater or equal than open parentheses 2 to the power of 1 third end exponent close parentheses cubed comma space i t space i s space c l e a r space t h a t comma space open parentheses 4 comma 2 to the power of 1 third end exponent close parentheses not an element of R T h u s comma space t h e space r e l a t i o n space i s space n o t space t r a n s i t i v e.

Answered by Vimala Ramamurthy | 3rd Mar, 2015, 12:24: PM