Download App

Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions3 Marks
Question
Determine whether the below relation is reflexive, symmetric and transitive:
Relation R in the set Z of all integers defined as
R = {(x, y) : x – y is an integer}
✓

Answer

It is given that Relation R in the set Z of all integers is defined as
R = {(x, y) : x – y is an integer}
Now, for every x $\in$ Z, (x, x) $\in$ R, as x - x = 0 is an integer.
$\Rightarrow$ R is reflexive.
Next, for every x, y $\in$ Z if (x, y) $\in$ R, then x - y is an integer.
$\Rightarrow$ -(x - y) is also an integer.
$\Rightarrow$ (y - x) is an integer.
$\Rightarrow$ (y - x) $\in$ R
$\Rightarrow$ R is symmetric.
Further, Take (x, y) ,(y, z) $\in$ R where x, y, z $\in$ R,
$\Rightarrow$ (x - y) and (y - z) are integers.
$\Rightarrow$ (x - z) = (x - y) + (y - x) is an integer.
$\Rightarrow$ (x, z) $\in$ R
$\Rightarrow$ R is transitive.
Therefore, R is reflexive, symmetric and transitive.

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

For the binary operation $\times _{10}$ on set $S = \{1, 3, 7, 9\},$ find the inverse of $3.$
→
Find the equation of the plane passing through the following point:
(-5, 0, -6), (-3, 10, -9) and (-2, 6, -6)
→
If the position vectors of the points A(3, 4), B(5, -6) and C(4, -1) are $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ respectively, compute $\vec{\text{a}}+2\vec{\text{b}}-3\vec{\text{c}}$.
→
Solve the following differential equations:
$\text{dy}+(\text{x}+1)(\text{y}+1)\text{dx}=0$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big)$
→
Show that $\text{y}=\text{ax}^3+\text{bx}^2+\text{c}$ is a solution of the differential equation $\frac{\text{d}^3\text{y}}{\text{dx}^3}=6\text{a}$
→
Find the intervals in which the function f given by $f(x) = 2x^2- 3x^2 - 36x + 7$ is $(a)$ strictly increasing, $(b)$ strictly decreasing.
→
Find the second order derivatives of the following functions:
$\text{y}=\text{x}^3\log\text{x}$
→
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 $\vec{\text{c}}=3\hat{\text{i}}-\hat{\text{k}}$
→
Find the mean and standard deviation of the following probability distributions:
$x_i$ -3 -1 0 1 3
$p_i$ 0.05 0.45 0.20 0.25 0.05
→