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

Show that ""^** : R xxR rarr R given by a^** b rarr a + 2b is not associative.

Show that the VV: RxxR rarr R given by (a, b) rarr max {a, b} and the ^^: RxxR rarr R given by (a, b) rarr min {a, b} are binary operations.

Show that + : Rxx R rarr R and xx: R xxR rarr R are commutative binary operations, but -: Rxx R rarr R and divid: R^(**)xxR^(**)rarrR^(**) are not commutative.

Show that **: RxxR rarr R defined by a^(**) b = a + 2b is not commutative.

Let P be the set of all subsets of a given set X. Show that cup: P xxP rarrP given by (A, B) rarr Acup B and cap : Pxx P rarrP given by (A, B) rarrr A cap B are binary operations on the set P.

Show that the function f : R rarr R given by f(x) =x^(3) is injective.

If f : R rarr R be given by f(x) = tan x , then f^(-1)(1) is .......

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.

Let A and B be sets. Show that f: A xxB rarr B xxA such that f(a,b) = (b,a) is bijecive function.

Prove that the function f :R rarr R , given by f(x) = 2x, is one-one and onto.