Prove that the operation `**` on `ZZ` defined by `a**b=a|b|` for all `a,binZZ` is a binary operation

Prove that the operation * an Z defined by a*b = a|b| for all a,b in Z is closed under*.

Examine whether the operation @ on ZZ^(+) defined by a@b=|a-b| for all a,binZZ^(+) , is a binary operation on ZZ^(+) or not.

Prove that the operation ^^ on RR defined by x^^y= min. of x and y for all x,yinRR is a binary operation on RR .

Prove that the binary operation ** on RR defined by a**b=a+b+ab for all a,binRR is commutative and associative.

The binary operation ** defined on NN by a**b=a+b+ab for all a,binNN is--

Let RR be the set of real numbers. Show that the operation ** defined on RR-{0] by a**b=|ab|,a,binRR-{0} is a binary operation on RR-{0}.

The binary operation * define on N by a*b = a+b+ab for all a,binN is

Let M_(2)={:[(x,0),(0,y)]:},x,yinRR-{0} be the set of 2xx2 matrices, prove that the operation ** defined on M_(2) by A**B=AB,A,BinM_(2) is a binary operation.

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.

Show that identity of the binary operation ** on RR defined by a**b=|a+b| for all a,binRR, does not exist.