Show that `**: R xx R to R` given by (a,b)` to a + 4b ^(2)` is a binary operation.

Show that ** R xxR to R given by a ** b to a + 2b is not associative.

Show that the vv:R xx R given by (a,b) to max {a,b} and the ^^ : R xx R to R given by (a,b) to min {a,b} are binary operations.

Show that **: R xx to R defined by a **b=a + 2b is not commutative.

Show that the operation ** defined on RR-{0} by a**b=|ab| is a binary operation. Show also that ** is commutative and associative.

Let P be the set of all subsets of a given set X. Show that uu:P xx P to P given by (A,B) to A uuB and nn:PxxP to P given by (A,B) to A nnB are binary operations on the set P.

Prove that the operation ** on ZZ defined by a**b=a|b| for all a,binZZ is a binary operation

Let @ be an operation defined on RR . The set of real numbers, by a@b=min(a,b) for all a,binRR . Show that @ is a binary operation on RR .

(I) Let ** be a binary operation defined by a**b=2a+b-3. Find 3**4. (ii) let ** be a binary operation on RR-{-1} , defined by a**b=(1)/(b+1) for all a,binRR-{-1} Show that ** is neither commutative nor associative. (iii) Let ** be a binary operation on the set QQ of all raional numbers, defined as a**b=(2a-b^(2)) for all a,binQQ . Find 3**5 and 5**3 . Is 3**5=5**3?

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a,b):a is sister of b} is the empty relation and R' ={(a,b): the difference between heights of a and b is less than 3 meters } is the universal relation.