Download App

Relations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRelations3 Marks
Question
Let A and B be two sets. Show that the sets A × B and B × A have elements in common iff the sets A and B have an elements in common.
✓

Answer

Let (a, b) be an arbitrary element of $(\text{A}\times\text{B})\cap(\text{B}\times\text{A}).$ Then,
$(\text{a},\text{b})\in(\text{A}\times\text{B})\cap(\text{B}\times\text{A})$
$\Leftrightarrow(\text{a},\text{b})\in\text{A}\times\text{B}$ and $(\text{a},\text{b})\in\text{B}\times\text{A}$
$\Leftrightarrow(\text{a}\in\text{A}\text{ and b}\in\text{B})$ and $(\text{a}\in\text{B and b}\in\text{A})$
$\Leftrightarrow(\text{a}\in\text{A and a}\in\text{B})$ and $(\text{b}\in\text{A and b}\in\text{B})$
$\Leftrightarrow\text{a}\in\text{A}\cap\text{B}$ and $\text{b}\in\text{A}\cap\text{B}$
Hence, the sets A × B and B × A have an element in comon iff the sets A and B have an element in common.

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 3 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

Evaluate the following limit: $\lim\limits_{\text{n}\rightarrow\infty}2^{\text{n}-1}\sin\Big(\frac{\text{a}}{2^\text{n}}\Big)$
→
The marks scored by Rohit in two tests were 65 and 70. Find the minimum marks he should score in the third test to have an average of at least 65 marks.
→
$\text{If}\ \text{cosec}\text{A}+\sec\text{A}=\text{cosecB}+\sec\text{B},$ prove that: $\tan\text{A}\tan\text{B}=\cot\frac{\text{A+B}}{2}$
→
$\frac{\text{c}-\text{b}\cos\text{A}}{\text{b}-\text{c}\cos\text{A}}=\frac{\cos\text{B}}{\cos\text{C}}$
→
Show that for any sets A and B,

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

→
Find the equation of the straight line passing through the point (2, 1) and bisecting the portion of the straight line 3x - 5y = 15 lying between the axes.
→
Let A and B be two stes such that: $\text{n(P)} = 20,$ $\text{n(A}\cup\text{B)=42 and n(A}\cap\text{B})=4.$ Find:
$\text{n(B)}.$
→
How many three-digit numbers are there, with distinct digits, with each digit odd?
→
Prove the following identities: $(\text{cosec}\text{x}−\sin\text{x})(\sec\text{x}−\cos\text{x})(\tan\text{x}+\cot\text{x})=1$
→
A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine
(i) P (not A) (ii) P (not B) and (iii) P (A or B)
→