question_answer 2)

Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a } is neither reflexive nor symmetric nor  transitive.

Answer:

R = {(a, b) : a  a, b, } Reflexive :       i.e.        (a, a)       Hence R is not reflexive. Symmetric : As 2, 5  and 2 < 25 i.e. 2 < 52.        (2, 5)       But 4 i.e.        but (5, 2)       Hence R is not symmetric.       Transitive : As 3, ?2 and ?1  and 3 < (2)2 and ?2 < (?1)2.        and (?2, ?1)       But                                      is not symmetric.       Transitive : (a, b)  and (b, c)        and              is transitive.                       Hence R is reflexive and transitive but not symmetric.