 Hello and welcome to the session it is called the following problem today given a non-empty set x consider the binary operation star from p of x cross p of x to p of x given by a star b is equal to a intersection b for all a b in p of x where p of x is the power set of x show that x is the identity element for this operation and x is the only invertible element in p of x with respect to the operation star now let us write the solution we have function star from p of x cross p of x to p of x defined by a star b which is equal to a intersection b for x belongs to p of x x star a is equal to x intersection a. Which is equal to a for all a thus x is the identity element now let i be another identity which implies i intersection a is equal to a for all a and if x belongs to x i intersection single term x is equal to single term x now x belongs to i which implies x is containing i and i is containing x which implies i is equal to x thus x is the only invertible element the identity element and x is the only invertible element i hope you understood this problem bye and have a nice day