Prove that the relation " friendship " is not an equivalence relation on the set of all people in Chennai.

Prove that set of similar triangles, 'is similar to' is an equivalence relation

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"} . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

Prove that the relation R defined on the set V of all vectors by vecaRvecbif veca=vecb, is an equivalence relation on V.

Show that the relation R in the set A of points in a plane given by R = {(P , Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set o

Let R be the realtion defined in the set A = {1,2,3,4,5,6,7} by R ={(a,b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1,3,5,7} are related to each other and all the elements of the subset {2,4,6} are related to each other, but no element of the subset {1,3,5,7} is related to any element of the subset {2,4,6}.

Let A = {a,b,c }. What is the equivalence relation of smallest cardinality on A ? What is the equivalence relation of largest cardinality on A?