Download App

Relations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRelations4 Marks
Question
Prove that:
$(\text{A}\cup\text{B})\times\text{C}=(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
✓

Answer

Let (a, b) bean arbitrary element of $(\text{A}\cup\text{B})\times\text{C}.$ Then,
$(\text{a},\text{b})\in(\text{A}\cup\text{B})\times\text{C}$
$\Rightarrow\text{a}\in\text{A}\cup\text{B}$ and $\text{b}\in\text{C}$ [By defination]
$\Rightarrow(\text{a}\in\text{A}\text{ or a}\in\text{B})$ and $\text{b}\in\text{C}$ [By defination]
$\Rightarrow(\text{a}\in\text{A}\text{ and b}\in\text{C})\text{ or }(\text{a}\in\text{B}\text{ and b}\in\text{C})$
$\Rightarrow(\text{a},\text{b})\in\text{A}\times\text{C or (a, b)}\in\text{B}\times\text{C}$
$\Rightarrow(\text{a},\text{b})\in(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
$\Rightarrow(\text{a, b})\in(\text{A}\cup\text{B})\times\text{C}$
$\Rightarrow(\text{a, b})\in(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
$\Rightarrow(\text{A}\cup\text{B})\times\text{C}\subseteq(\text{A}\times\text{C})\cup(\text{B}\times\text{C})\ ...(\text{i})$
Again, let (x, y) be an abitrary element of $(\text{A}\times\text{C})\cup(\text{B}\times\text{C}).$ Then,
$(\text{x, y})\in(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
$\Rightarrow(\text{x, y)}\in\text{A}\times\text{C}\text{ or (x, y)}\in\text{B}\times\text{C}$
$\Rightarrow\text{x}\in\text{A and y}\in\text{C or x}\in\text{B and y}\in\text{C}$
$\Rightarrow(\text{x}\in\text{A or x}\in\text{B})\text{ and y}\in\text{C}$
$\Rightarrow\text{x}\in\text{A}\cup\text{B and y}\in\text{C}$
$\Rightarrow(\text{x},\text{y})\in(\text{A}\cup\text{B})\times\text{C}$
$\Rightarrow(\text{x},\text{y})\in(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
$\Rightarrow(\text{x, y})\in(\text{A}\cup\text{B})\times\text{C}$
$\Rightarrow(\text{A}\times\text{C})\cup(\text{B}\times\text{C})\subseteq(\text{A}\cup\text{B})\times\text{C}\ ...(\text{ii})$
Using equation (i) and equation (ii), we get
$(\text{A}\cup\text{B})\times\text{C}=(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
Hence 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 "(Each question 4 marks)" from Relations→Relations — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the sum of the following series to n terms: $1^3 + 3^3 + 5^3 + 7^3 + ......$
→
Find the equation of a line which passes through the point (22, -6) and is such that the intercept of x-axis exceeds the intercept of y-axis by 5.
→
Prove that: $\sin3\text{x}+\sin2\text{x}-\sin\text{x}=4\sin\text{x}\cos\Big(\frac{\text{x}}{2}\Big)\cos\Big(\frac{3\text{x}}{2}\Big)$
→
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
→
Differentiate the following from first principle$\tan\text{2x}$
→
Find the equation of the circle which circumscribes the triangle formed by the lines $y = x + 2, 3y = 4x$ and $2y = 3x$.
→
For any two sets of A and B, prove that:

$\text{A}'\cup\text{B}=\text{U}\Rightarrow\text{A}\subset\text{B.}$

→
If a, b, c are in A.P., prove that the straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.
→
Prove the following identities: $\text{cosec}\text{ x}(\sec\text{x}−1)−\cot\text{x}(1−\cos\text{x})=\tan\text{x}−\sin\text{x}$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\text{x}-\sin2\text{x}}{\text{x}^3}$
→