Let S R x R and consider the relation x1x2Ry1y2 if and only

Let S= R x R and consider the relation (x1,x2)R(y1,y2) if and only if x1^2+x2^2=y1^2+y2^2. Prove or disprove that R is an equivalence relation.

Solution

1.

x1^1+x2^2=x1^2+x2^2

HEnce, R is reflexive

2.

x1^1+x2^2=y1^2+y2^2

implies

y1^2+y2^2=x1^1+x2^2

Hence, R is symmetric

3.

x1^1+x2^2=y1^2+y2^2 and y1^2+y2^2=z1^1+z2^2

implies

x1^1+x2^2=z1^2+z2^2

Hence, R is transitive.

Hence R is an equivalence relation.