Let A be the set of all functions f: Z Z and R be a relation on A such that R = ( f , g ): f(0)= g (1) and f(1)= g (0) . Then R is:

Step-by-step solution

aligned & R = ( f , g ): f (0)= g (1) and f (1)= g (0) & Reflexive: ( f , f ) R & = f (0)= f (1) and f (1)= f (0) must hold & but this is not true for all function aligned so not reflexive Symmetric: If ( f , g ) R ( g , f ) R Now, g(0)=f(1) and g(1)=f(0) true symmetric Transitive : If ( f , g ) R and ( g , h ) R ( f , h ) R Now (f, g) R f(0)=g(1) and f(1)=g(0) ( g , h ) R g (0)= h (1) and g (1)= h (0) For (f, h) R we need f(0)=h(1) and f(1)=h(0) Now f(0)=g(1)=h(0) and f(1)=g(0)=h(1) Hence not transitive