Consider the binary operation ^* and o defined by the following t… - Vidyadip

Question

Consider the binary operation $^*$ and $o$ defined by the following tables on set $S = \{a, b, c, d\}.$

$*$	$a$	$b$	$c$	$d$
$a$	$a$	$b$	$c$	$d$
$b$	$b$	$a$	$d$	$c$
$c$	$c$	$d$	$a$	$b$
$d$	$d$	$c$	$b$	$a$

✓

Answer

Commutativity:The table is symmetrical about the leading element. It means * is commutative on $S.$
Associativity:
$a * (b * c) = a * d$
$= d$
$(a * b) * c = b * c$
$=d$
Therefore,
$a * (b * c) = (a * b) * c \forall\text{ a, b, c}\in\text{S}$
So, * is Associative on $S.$
Finding identity element:
We observe that the first row of the composition table coincides with the top$-$most row and the first column coincides with the left$-$most column.
These two intersect at a.
$\Rightarrow x * a = a * x = x, \forall\text{ x}\in\text{S}$
So, a is the identity element:
$a * a = a$
$\Rightarrow a^{-1} = a$
$b * b = a$
$\Rightarrow b^{-1} = b$
$c * c = a$
$\Rightarrow c^{-1} = c$
$d * d = a$
$\Rightarrow d^{-1} = d$

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 "5 Marks Questions" from Binary Operations→Binary Operations — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

By computing the shortest distance determine whether the following pairs of lines intersect or not:3
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{-5}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-7}{1}=\frac{\text{z}-5}{3}$

Three schools A, B and C organised a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of ₹ 25, ₹ 100 and ₹ 50 each. The number of articles sold are given below:

School	A	B	C
Article	A	B	C
Hand - fans	40	25	35
Mats	50	40	50
Plates	20	30	40

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation.

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

Find one-parameter families of solution curves of the following differential equation: $($or solve the following differential equation$)\frac{\text{dy}}{\text{dx}}+3\text{y}=\text{e}^{\text{mx}}, m$ is given real number.

If $\text{xy}=4,$ prove that $\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=3\text{y}$

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Let A = {a, b, c}, B = {u, v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as:
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.

Solve the following differential equations:$\cos\text{y}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}},\text{y}(0)=\frac{\pi}{2}$

The two adjacent sides of a parallelogram are $2\hat{i} - 4\hat{j} - 5\hat{k} \text{ and } 2\hat{i} + 2\hat{j} + 3\hat{k}.$ Find the two unit vectors parallel its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Show that the following system of linear equations is consistent and also find solution:
$2x + 3y = 5$
$6x + 9y = 15$