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 for all a, b ∈ Q.

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q.}$ Then,
a * b = a - b
b * a = b - a
Therefore,
$\text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
Thus, * is not commutative on Q.
Associativity: Let $\text{a, b, c}\in\text{Q.}$ Then,
a * (b * c) = a * (b - c)
= a - (b - c)
= a - b + c
(a * c) * c = (a - b) * c
= a - b - c
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 "Solve the Following Question.(5 Marks)" from Binary Operations→Binary Operations — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following definite integrals:$\int_{0}^\limits{2}\frac{1}{4+\text{x}-\text{x}^2}\text{ dx}$

At what points on the following curves, is the tangent parallel to x-axis?
$\text{y}=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$

verify that $\text{y}=\log(\text{x}+\sqrt{\text{x}^2+\text{a}^2})^2$ is a solution of the differential equation $(\text{a}^2+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}=0$

Find the second order derivatives of the following functions:
$\text{y}=\tan^{-1}\text{x}$

Two dice are thrown together and the total score is noted. The event E, F and G are "a total of 4", "a total of 9 or more", and "a total divisible by 5", respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=(\sin2\text{x}+\cot\text{x}+2)^2\text{at}\text{ x}=\frac{\pi}{2}$

The probabilities of two students A and B coming to the school in time are $\frac{3}{7}$ and $\frac{5}{7}$ respectively. Assuming that the events, 'A coming in time' and 'B coming in time' are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.

Two cards are selected at random from a box which contains five cards numbered $1, 1, 2, 2,$ and $3$. Let X denote the sum and Y the maximum of the two numbers drawn. Find the probability distribution, mean and variance of X and Y.

Evaluate the following integrals:
$\int\cos^{-1}(4\text{x}^3-3\text{x})\text{dx}$

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.