Define an operation * on the set of all relational numbersas follows:

r * s = r + s − (r × s),for any two rational numbers r, s. Answer the following with justification:

(i) Is closed under the operation ?

(ii) Is an associative operation on ?

(iii) Is * a commutative operation on ?

(iv) What is a * 1 for any a in

(v) Find two integers a ≠ 0 and b ≠ 0 such that a * b = 0.

Open in App

Solution

We are given an operation on the set of all rational numbers as:

(i)

The set of all rational numbers is closed under addition and multiplication.

Thus, is closed under the operation .

(ii)

Since, the operation is an associative operation on .

(iii)

Since, the operation * is a commutative operation on .

(iv)

a*1 = 1 for any a in

(v)

It is true for .

Suggest Corrections

Similar questions

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b $-$ ab, for all a, b $\in$ S.
Prove that:
(i) * is a binary operation on S
(ii) * is commutative as well as associative. [CBSE 2014]

Determine whether the relation R defined on the set R of all real numbers as R = {(a, b) ; a, b $ϵ$ R and a-b + $\sqrt{3} ϵ$ S, where S is the set of all irrational numbers}, is reflexive, symmetric and transitive.

Let A = $R \times R$ and $*$ be the binary operation on A defined by (a,b) $*$ (c, d) - (a + c, b + d).

Prove that $*$ is commutative and associative. Find the identity element for $*$ on A. Also write the inverse element of the element (3, -5) in A.

Q. Consider the binary operations*: R × R → and o: R × R → R defined as and a o b = a , &mnForE; a , b ∈ R . Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE; a , b , c ∈ R , a *( b o c ) = ( a * b ) o ( a * c ). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Q. Let * be a binary operation on Q₀ (set of non-zero rational numbers) defined by

a * b = \frac{a b}{5} for all a, b \in Q_{0}

.
Show that * is commutative as well as associative. Also, find its identity element if it exists.

Q. Let S be the set of all real numbers except −1 and let '*' be an operation defined by

a * b = a + b + ab for all a, b ∈ S.

Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.

Join BYJU'S Learning Program