Download App

RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Let n be a fixed positive integer. Define a relation R in Z as follows $\forall\ \text{a},\ \text{b}\in\text{Z},$ aRb if and only if a - b is divisible by n. Show that R is an equivalance relation.
✓

Answer

Given that, $\forall\ \text{a},\ \text{b}\in\text{Z},$ aRb if and only if a - b is divisible by n. Now, I. Reflexive aRa ⇒ (a - a) is divisible by n, which is true for any integer 'a' as ‘0’ is divisible by n.Hence, R is reflexive.
II. Symmetric.
aRb ⇒ a - b is divisible by n. ⇒ -(b - a) is divisible by n. ⇒ (b - a) is divisible by n. ⇒ bRaHence, R is symmetric.
III. Transitive.
Let aRb and bRc
⇒ (a - b) is divisible by n and (b - c) is divisible by n. ⇒ (a - b) + (b - c) is divisible by n. ⇒ (a - c) is divisible by n. ⇒ aRcHence, R is transitive.
So, R is an equivalence relation.

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Consider the following Linear Programming Problem:
Minimise Z = x + 2y
Subject to $2 x+y \geq 3, x+2 y \geq 6, x, y \geq 0$
Show graphically that the minimum of Z occurs at more than two points
→
A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10cm/ sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall.
→
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b - 4. Prove that * is neither commutative nor associative on Z.
→
Find $\big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\big]$, when
$\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}},\vec{\text{b}}=\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}-\hat{\text{k}}$
→
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog.
→
Find the area of the parallelogram determinrd by the vectors:
$2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}$ and $\hat{\text{i}}-\hat{\text{j}}$
→
Integrate the function: $\frac{1}{\cos ^{2} x(1-\tan x)^{2}}$
→
Let f be an invertible real function. Write $\mathrm{(f^{-1}of)(1) + (f^{-1}of)(2) + ..... + (f^{-1}of)(100).}$
→
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
$(0.0037)^{\frac{1}{2}}$
→
Differentiate the following functions with respect to x:
$3^{\text{x}\log\text{x}}$
→