Show that for any sets A and B, A=(A ) (A - B) - Vidyadip

Question

Show that for any sets A and B,

$\text{A}=(\text{A}\cap\text{B})\cap(\text{A - B})$

✓

Answer

We know that $(\text{A}\cap\text{B})\subset\text{A and (A - B)}\subset\text{A}$
$\Rightarrow(\text{A}\cap\text{B})\cap(\text{A - B})\subset\text{A}......(\text{i})$
Let and $\text{x}\in(\text{A}\cap\text{B})\cap(\text{A - B})$
$\Rightarrow\text{x}\in(\text{A}\cap\text{B})\text{ and x}\in(\text{A}-\text{B})$
$\Rightarrow\text{x}\in\text{A and}\text{ x}\in\text{B and }\text{x}\in\text{A and }\text{x}\not\in\text{B}$
$\Rightarrow\text{x}\in\text{A and }\text{x}\in\text{A}$ [$\because\text{ x}\in\text{A}\text{ x}\not\in\text{A}$ are not possible simutaneusly]
$\Rightarrow\text{x}\in\text{A}$
$\therefore(\text{A}\cap\text{B})\cap\text{(A - B)}\subset\text{A}..........\text{(ii)}$
From (i) and (ii), we get
$\text{A}=(\text{A}\cap\text{B})\cap(\text{A - B}).$

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 Sets→Sets — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\text{a}\in[-1,2,3,4]$ and $\text{b}\in [0, 3, 6],$ write the set of all ordered pairs (a, b) such that a + b= 5.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\tan\text{x}}{1-\cos\text{x}}$

Let R be a relation from N to N defined by $\text{R}=\{(\text{a, b}):\text{a, b}\in\text{N and a}=\text{b}^2\}.$ Are the following statement true?
$(\text{a, b})\in\text{R }\Rightarrow \text{(b, a)}\in\text{R}$

Planes are drawn parallel to the coordinate planes through the points (3, 0, -1) and (-2, 5, 4).
Find the lengths of the edges of the parallelopiped so formed.

Find the middle term in the expansion of:

$\Big(\frac{2}{3}\text{x}-\frac{3}{2\text{x}}\Big)^{20}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}(\cos\text{x}+\text{a}\sin\text{bx})^\frac{1}{\text{x}}$

Mean and standard deviation of 100 items are 50 and 4, respectively. Find the sum of all the item and the sum of the squares of the items.

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the path traced by the man.

A farmer buys a used tractor for ₹12000. He pays ₹ 6000 cash and agrees to pay the balance in annual installments of ₹ 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?

Solve the inequality and represent the solution graphically on number line: 2 (x – 1) < x + 5, 3 (x + 2) > 2 – x