Short Answer TypeShow 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 *?
Here * is a binary operation on N given by a * b = l.c.m. (a, b), a, b ∈ N
(i) 5 * 7 = 35, 20 * 16 = 80
(ii) Since l.c.m. (m, n) = l.c.m. (n, m) ∴ m * n = n * m ∀ m, n ∈ N binary operation is commutative
(iii) Let a, b, c ∈ N
Now a * (b * c) = l.c.m. (a, b * c)
= l.c.m. [a, l.c.m (b. c)]
= l.c.m [l.c.m. (a, b), c]
= l c.m. [(a * b), c]
∴ a * (b * c) = (a * b) * c ∀ a, b, e ∈ N binary operation is associative.
(iv) Let e be identity element. Then
V
a * e = a = e * a ∀ a ∈ N
⇒ (a * e) = a ∀ a ∈N
⇒ l.c.m. (a, e) = a ∀ a ∈ N
⇒ e = 1
∴ 1 is the identity element in N
(v) Let a be an invertible element in N.
Then there exists such that
a * b = 1 ⇒ l.c.m. (a, b) = 1 ⇒ a = b = 1
∴ 1 is the invertible element of N.
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