Question

Let M be the set of all functions from N to N . Let us define a relation R on M as follows, R = { ( f , g ) ∈ M × M : f ( 2 ) = g ( 2 ) or f ( 10 ) = g ( 10 )

Answer 1

Given M,

a set of functions on the set of natural numbers N .

In the set M, a relation R is defined as follows;

for f, g ∈ M, fRg if f(2) = g(2) or f(10) = g(10) .

Clearly, we have ;

(I) for each f in M, fRf because; f(2) = f(2) or f(10) = f(10).

Therefore, R is reflexive (II) let fRg ==> f(2) = g(2) or f10) = g(10) ==> g(2) = f(2) or g(10) = f(10) ==> gRf .

Therefore, R is symmetric .

(III) R is transitive. For that, let fRg ==> f(2) = g(2) or f(10) = g(10);

further assume that gRh then g(2) = h(2) or g(10) = h(10) .

Then, from above, it is obvious that f(2) = g(2) = h(2) or

f(10) = g(10) = h(10) ==> fRh .

Hence R is an equivalence relation