Download App

RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS3 Marks
Question
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by $\text{A}*\text{B}=\text{A}\cap\text{B}\ \ \forall\ \text{A},\text{ B}$ in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation *.
✓

Answer

It is given that *: P(X) × P(X) → P(X) is defined as $\text{A}*\text{B}=\text{A}\cap\text{B}\ \ \forall\ \text{A},\text{ B}\in\text{P(X)}$
We know that $\text{A}\cap\text{X}=\text{A}=\text{X}\cap\text{A}\forall\text{A}\in\text{P(X)}$
Thus, X is the identity element for the given binary operation *.
Now, an element $\text{A}\in\text{P(X)}$ is invertibleif there exists $\text{B}\in\text{P(X)}$ such that
A * B = X = B * A. (As X is the identity element)
i.e.,
$\text{A}\cap\text{B}=\text{X}=\text{B}\cap\text{A}$
This case is possible only when A = X = B.
Thus, X is the only invertible element in P(X) with respect to the given operation *.
Hence, the given result is proved.

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 RELATIONS AND FUNCTIONS→RELATIONS AND FUNCTIONS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the angle between the lines $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{{x - 5}}{4} = \frac{{y - 2}}{1} = \frac{{z - 3}}{8}$.
→
Prove that $f(x) = ax + b,$ where $a, b$ are constants and $a < 0$ is an decreasing function on $R.$
→
Find the identity element in the set of all rational numbers except -1 with respect to * defined by a * b = a+ b + ab.
→
Find the angle between the given planes.
$\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}})=1$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}+\hat{\text{j}})=4$
→
If the mean of a binomial distribution is 20 and its standard deviation is 4, find p.
→
Find the particular solution of the differential equation $(1+\text{e}^{2\text{x}})\text{dy}+(1+\text{y}^2)\text{e}^{\text{x}}\ \text{dx}=0,\ \text{given that y}=1\ \text{when x}=0. $
→
Differentiate the following functions with respect to x:
$3^{\text{x}^2+2\text{x}}$
→
If A, B, C, D are the points with position vectors $\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}},\ 2\hat{\text{i}}-3\hat{\text{k}},$ respectively, find the projection of $\overrightarrow{\text{AB}}$ along $\overrightarrow{\text{CD}}.$
→
Show that $|\vec{a}| \vec{b}+|\vec{b}| \vec{a}$ is perpendicular to $|\vec{a}| \vec{b}-|\vec{b}| \vec{a}$ for any two nonzero vectors $\vec a$ and $\vec b$. 
→
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q.
→