Define an operation * on Q as follows: `a * b =((a+b)/(2)), a, b in Q`. Examine the existence of identify and existence of inverse for the operation * on Q.

Verify the Existence of inverse for the operation +_(4)" on "Z_(4) .

Define an operation * on Q as follows: a * b =((a+b)/(2)), a, b in Q . Examine the closure, communative, and associative properties satisfied by * on Q.

Define *" on Q as " a * b =((2a+b)/(2)), a , b in Q . Then the identify element is :

Let M={[{:(,x,x),(,x,x):}}: x in R-{0}} and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the existence of identify, existence of inverse properties for the operation * on M.

(i) Let M={({:(x,x),(x,x):}):x in R -{0}} and let ** be the matrix multiplication. Determine whether M is closed under ** . If so, examinie the existence of identity, existence of inverse properties for the operation ** on M.

Let A be Q/{1}. Define * on A by x * y=x+y-xy. Is * binary on A? If so, examine the existence of identity & inverse properties for the operation * on A.