293. Show that * : R x R → R given by a * b → a + 2 b is not associative.

Show that zero is the identity for addition on R and 1 is the identity for multiplication on R. But there is no identity element for the operations
– : R x R → R and ÷ : R_* x R_* → R_*.

Show that – a is not the inverse of a ∈ N for the addition operation + on N and  is not the inverse of a ∈ N for multiplication operation x on N, for a ≠1.

Let * be a binary operation on the set Q of rational numbers given as a * b = (2a – b)², a, b ∈ Q. Find 3 * 5 and 5 * 3. Is 3 * 5 = 5 * 3? 

Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this

(i) On Z⁺, define * by a * b = a – b (ii) On Z⁺, define *by a * b = a b
(iii) On R , define * by a * b = a b² (iv) On Z⁺, define * by a * b = | a – b |
(v) On Z⁺ , define * by a * b = a

Let * be a binary operation on the set Q of rational numbers as follows :

(i) a * b = a – b    (ii) a * b = a² + b²

Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find

(i) 5 * 7, 20 * 16    (ii) Is * commutative?
(iii) Is * associative?    (iv) Find the identity of * in N.
(v) Which elements of N are invertible for the operation *?