Given a non-empty set X, Let *: `P(x) xx P(x)` be defined as `A**B =(A-B) cup (B-A), AA A,B in P(x)`. Show that the empty set `phi` is the identity for the operation * and all the elements A of p(x) are invertible with `A^(-1)=A`.

Show that the function e^x/x^p is strictly increasing for x gt p gt 0 .

If A be non-empty set of children in a family then the relation "a is a brother of b" on A is-

A binary operation * on the set {0,1,2,3,4,5} is defined as a*b= {{:(a+b, "if" a+b lt 6),(a+b-6, "if" a +b gt=6):} Find the composition table for * Also, show that zero is the identity for this operation and each non-zero element a of the set is invertible with 6-a, being the inverse of a.

Let S =Q xx Q and let * be a binary operation on S defined by (a, b)* (c,d) = (ac, b+ad) for (a, b), (c,d) in S. Find the identity element in S and the invertible elements of S.

If P(A)=0.6, P(B | A)= 0.5, find P (A cup B) if A, B are independent.

Let R^(+) be the set of all positive real numbers. If f: R^(+) rarr R^(+) is defined as f(x)= e^(x), AA x in R^(+) , then check whether f is invertible or not.

Let a relation R on the set A of real numbers be defined as (a, b) inR implies1+ab >0, AA a, b in A . Show that R is reflexive and symmetric but not transitive.

Let f : W rarr W be defined as f (x) = x -1 if x is odd and f(x) = x +1 if x is even then show that f is invertible. Find the inverse of f where W is the set of all whole numbers.