Test whether the following relations R_2 are: 1. Reflexive. 2. Sy… - Vidyadip

Question

Test whether the following relations $R_2$ are:

Reflexive.
Symmetric.
Transitive.

$R_2$ on Z defined by $(\text{a, b})\in\text{R}_2\Leftrightarrow\ |\text{a}-\text{b}|\leq5$

✓

Answer

Reflexivity: Let a be an arbitrary element of $R_2$. Then,$\text{a}\in\text{R}_2$
$\Rightarrow\ |\text{a}-\text{a}|=0\leq5$
So, $R_2$ is reflexive.
Symmetry: Let $(\text{a, b})\in\text{R}_2$
$\Rightarrow\ |\text{a}-\text{b}|\leq5$
$\Rightarrow\ |\text{b}-\text{a}|\leq5$ [Since, |a - b| = |b - a|]
$\Rightarrow\ (\text{b, a})\in\text{R}_2$
So, $R_2$ is symmetric.
Transitivity: Let $(1, 3)\in\text{R}_2$ and $(3,7)\in\text{R}_2$
$\Rightarrow\ |1-3|\leq5$ and $|3-7|\leq5$
But $|1-7|\nleq5$
$\Rightarrow\ (1,7)\notin\text{R}_2$
So, $R_2$ is 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

Solve the following equation for x:
$\tan^{-1}\frac{\text{x}-2}{\text{x}-1}+\tan^{-1}\frac{\text{x}+2}{\text{x}+1}=\frac{\pi}{4}$

By using properties of determinants, show that:
$\begin{vmatrix}x+4&2x&2x\\2x&x+4&2x\\2x&2x&x+4\end{vmatrix}=(5x+4)(4-x)^2$

A particle moves along the curve $y = x^3. $ Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-coordinate.

Evaluate the following integrals:
$\int\text{x e}^{\text{x}^2}=\text{dx}$

Evaluate the following integrals:
$\int\text{x}\sin2\text{x dx}$

Let $R_0 $ denote the set of all non-zero real numbers and let $A = R_0 \times R_0.$ If '*' is a binary operation on adefined by,
$(a, b) * (c, d) = (ac, bd)$ for all $(a, b), (c, d) \in A$
Show that $'*'$ is both commutative and associative on $A.$

Define $\vec{\text{a}}\times\vec{\text{b}}$ and prove that $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\big(\vec{\text{a}}.\vec{\text{b}}\big)\tan\theta,$ where $\theta$ is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$

Given a non-empty set $X,$ let $*: P(X) \times P(X) \rightarrow P(X)$ be defined as $\text{A}*\text{B} =(\text{A – B})\cup(\text{B – A}),\forall\text{A},\text{B}\in\text{P(X)}.$Show that the empty set $\phi$ is the identity for the operation $*$ and all the elements $A$ of $P(X)$ are invertible with $A^{–1} = A. (\text{Hint: }(\text{A}-\phi)\cup(\phi-\text{A})=\text{A}\ \text{and }(\text{A}-\text{A})\cup(\text{A}-\text{A})=\text{A}*\text{A}=\phi).$

Solve the following linear programming problem graphically:
Maximise Z = 4x + y subject to the constraints:
x + y $\le$ 50
3x + y $\le$ 90
x $\ge$ 0, y $\ge$ 0

Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is