Inverse of an element

Prove that the operaton ** on QQ-{1} given by a*b=a+b-ab for all a,binQQ-{1} (i) is closed: (ii) satisfies the commutative and associative laws, (iii) Find the identity element, (iv) Find the inverse of any element ainQQ-{1}.

Show that the operation ** on ZZ , the set of integers, defined by. a**b=a+b-2 for all a,b inZZ (i) is a binary operation: (ii) satisfies commutaitve and associative laws: (iii) Find the identity elemetn in ZZ , (iv) Also find the inverse of an element ainZZ.

The binary operation multiplication modulo 10(xx_(10)) is defined on the set A={0,1,3,7,9,}, find the inverse of the element 7.

A binary operation * is defined on the set X=R-{-1} by x**y=x+y +xy, AA x, y in X . Check whether * is commutative and associative. Find its identity element and also find the inverse of each element of X.

If +_(6) (addition modulo 6) is a binary operation on A={0,1,2,3,4,5}, find the value of 3+_(6)3^(-1)+_(6)2^(_1) . [note that the identity element is 0 and the inverse of the element 2 is 4 as 2+_(6)4=0 , the identity element.]

Consider the binary operation * and o defined by the following tables on set S={a,b,c,d}.( FIGURE) Show that both the binnary operations are commutative and associative.Write down the identities and list the inverse of elements.

Consider the binary operation * and o defined by the following tables on set S={a ,\ b ,\ c ,\ d} . (FIGURE) Show that both the binary operations are commutative and associative. Write down the identities and list the inverse of elements.

Let A=QxxQ , where Q is the set of all rational numbers and '**' be the operation on A defined by : (a,b)**(c,d)=(ac,b+ad)" for "(a,b),(c,d)inA . Then, find : (i) The identity element of '**' in A (ii) Invertible elements of A and hence write the inverse of elements (5, 3) and ((1)/(2),4) .