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.

For the binary operation * on Z defined by a*b= a+b+1 the identity element is

The binary operation defined on the set z of all integers as a ** b = |a-b| - 1 is

Let ** be a binary operation on the set Q of rational numbers as follows: (i) a**b=a-b (ii) a**b=a^2+b^2 Find which of the binary operations are commutative and which are associative

Show that the binary operation * on Z defined by a*b= 3a+7b is not commutative.

Define a commutative binary operation on a set.

Find the number of binary operations that can be defined on the set A={a,b,c}

A binary operation is chosen at random from the set of all binary operations on a set A containing n elements. The probability that the binary operation is commutative, is

Let * be a binary operation on the set Q_0 of all non-zero rational numbers defined by a*b= (a b)/2 , for all a ,\ b in Q_0 . Show that (i) * is both commutative and associative (ii) Find the identity element in Q_0 (iii) Find the invertible elements of Q_0 .

Consider the binary operations *: RxxR->R and o: RxxR->R defined as a*b=|a-b| and aob=a for all a ,\ b in Rdot Show that * is commutative but not associative, o is associative but not commutative.

Examine whether the binary operation ** defined on R by a**b=a b+1 is commutative or not.