14. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

The given relation in the set L= all lines in XY− plane is defined asR={(L1,L2):L1 is parallel to L2}Let L1∈A then as L1 is parallel to L1 ,(L1,L1)∈RSo, R is reflexiveLet L1,L2∈A and (L1,L1)∈RThen, L1 is parallel to L2L2 is parallel to L1So, (L2,L1)∈Ri.e., R is symmetricLet L1,L2,L3∈A and (L1,L2) and (L2,L3)∈RThen, L1?L2 and L2?L3So, L1?L3i.e., (L1,L2)∈RSo, R is transitiveHence, R is an equivalence relationThe set of lines related to y=2x+4 is given by the equation y=2x+C where C is some constant.

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Relations and Functions Ncert Solutions Maths class 12th Maths Class 12th Relations and Functions

14. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L₁, L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

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Vishal Baghel | Contributor-Level 10

4 months ago

The given relation in the set $L =$ all lines in $X Y -$ plane is defined as

$R = {(L_{1}, L_{2}) : L_{1}$ is parallel to $L_{2}}$

Let $L_{1} \in A$ then as $L_{1}$ is parallel to $L_{1}$ ,

$(L_{1}, L_{1}) \in R$

So, $R$ is reflexive

Let $L_{1}, L_{2} \in A$ and $(L_{1}, L_{1}) \in R$

Then, $L_{1}$ is parallel to $L_{2}$

$L_{2}$ is parallel to $L_{1}$

So, $(L_{2}, L_{1}) \in R$

i.e., $R$ is symmetric

Let $L_{1}, L_{2}, L_{3} \in A$ and $(L_{1}, L_{2})$ and $(L_{2}, L_{3}) \in R$

Then, $L_{1} ? L_{2}$ and $L_{2} ? L_{3}$

So, $L_{1} ? L_{3}$

i.e., $(L_{1}, L_{2}) \in R$

So, $R$ is transitive

Hence, $R$ is an equivalence relation

The set of lines related to $y = 2 x + 4$ is given by the equation $y = 2 x + C$ where $C$ is some constant.

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Let S = {1, 2, 3 ,....,20}

R₁ = {(a, b) : a divide b}

R₂ = {(a, b) : a is integral multiple of b} a, b Î s

n(R₁ – R₂) = ?

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R₁ = { (1, 1) (1, 2), (1, 3)., (1, 20), (2, 2), (2, 4). (2, 20), (3, 3), (3, 6), . (3, 18),
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If A = {1, 2, 3, … 100}, R = {(x, y) | 2x = 3y, x, y ∈ A} is symmetric relation on A and the number of elements in R is n, the smallest integer value of n is

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$?$ R is symmetric relation

⇒ (y, x) ∈ R V (x, y) ∈ R

(x, y) ∈ R ⇒ 2x = 3y and (y, x) ∈ R ⇒ 3x = 2y

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f is increasing function

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$\underset{x \to \infty}{l i m} \frac{f (x)}{f (x)} < \underset{x \to \infty}{l i m} \frac{f (5 x)}{f (x)} < \underset{x \to \infty}{l i m} \frac{f (7 x)}{f (x)}$

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Given f (k) = ${\begin{cases} k + 1, k i s o d d \\ k, k i s e v e n \end{cases}$

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So total no. of functions = 10⁵ × 1 = 10⁵

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14. Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L₁, L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

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