Show that the operation * given by x*y=x+y+ -xy is a binary oeration on Z,Q and R but not on N.

Show that * On Z^(+) defined by a*b = |a-b| is a binary operation or not.

Show that * On R defined by a*b = a +4 b^(2) is a binary operation or not.

If R={(x, y): x +2y =8} is a relation on N, then write the range of R.

Show that addition, subtraction and multiplication are binary operations on R. Also, show that division is not a binary operation on the set R.

Determine whether a** b =ma - nb "on" Q + "where" m "and" n in N operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

Let. X = {1, 2, 3, 4, 5, 6, 7, 8,9}. Let R_(1) , be a relation on X given by R_(1) ={(x, y): x - y is divisible by 3)} and R_(2) , be another relation on X given by R_(2) ={(x, y): {x,y) sub (1,4,7) or (x, y) sub (2,5,8) or (x, y) sub (3, 6, 9)}. Show that R_(1)=R_(2) .

Show that the relation R defined on the set Z of all integers defined as R={(x,y):x-y is an integer} is reflexive, symmertric and transtive.

Show that the family of curves for which the slope of the tangent at any point (x, y) on it is (x^(2)+y^(2))/(2xy) , is given by x^(2)-y^(2)= Cx .

If f:X to Y is a function. Define a relation R on X given by R={(a, b): f(a)=f(b)}. Show that R is an equivalence relation on X.