Check the commutativity and associativity of the following binary… - Vidyadip

Question

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $a * b = (a - b)^2$ for all $a, b \in Q.$

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}.$ Then,$a * b = (a - b)^2$
$= (b - a)^2$
$= b * a$
Therefore,
a * b = b * a, $\forall\ \text{a, b}\in\text{Q}$
Thus, * is commutative on Q.
Associativity: Let $\text{a, b, c}\in\text{Q}.$ Then,
$a * (b * c) = a * (b - c)^2$
$= a * (b^2 + c^2 - 2bc)$
$= (a - b^2 - c^2 + 2bc)^2$
$(a * b) * c = (a - b)^2 * c$
$= (a^2 + b^2 - 2ab) * c$
$= (a^2 + b^2 - 2ab - c)^2$
Therefore,
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})\neq(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}$
Thus, * is not associative on Q.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Binary Operations→Binary Operations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\sin^{-1}\Big(\frac{2\tan\text{x}}{1+\tan^2\text{x}}\Big)\text{dx}$

Let f be an invertible real function. Write $\mathrm{(f^{-1}of)(1) + (f^{-1}of)(2) + ..... + (f^{-1}of)(100).}$

Express the matrix $\begin{bmatrix} 0 & \frac{9}{2} & \frac{9}{2} \\ -\frac{9}{2} & 0 & -\frac{3}{2} \\ -\frac{9}{2} & \frac{3}{2} & 0 \end{bmatrix} $ as the sum of a symmetric and skew symmetric matrix.

Find $\frac{\text{dy}}{\text{dx}}$ of the functions expressed in parametric:
$\sin\text{x}=\frac{2\text{t}}{1+\text{t}},\ \tan\text{y}=\frac{2\text{t}}{1-\text{t}^2}.$

Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.

Find the magnitude of $\vec{\text{a}}=\big(3\hat{\text{k}}+4\hat{\text{j}}\big)\times(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}).$

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

Obtain the equation of the plane passing through the point (1, - 3, -2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.

Urn $A$ contains $1$ white, $2$ black and $3$ red balls; urn $B$ contains $2$ white, $1$ black and $1$ red ball; and urn $C$ contains $4$ white, $5$ black and $3$ red balls. One urn is chosen at random and two balls are drawn. These happen to be one white and one red. What is the probability that they come from urn $A$?

Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.