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

10. Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

19 Views | Posted 4 months ago

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Answered by

Vishal Baghel | Contributor-Level 10

4 months ago

Let A= ${a, b, c}$

(i) R= ${(a, b), (b, a)}$ is a relation in set A

So, $(a, b) \in R$ and $(b, a) \in R \to$ Symmetric

$(a, a) \notin R \to$ not reflexive

$(a, b) \in R, (b, a) \in R$ but $(a, a) \notin R$ $\to$ not transitive

(ii) R= ${(a, b), (b, c), (a, c)}$ is a relation in set A

So, $(a, a) \notin R \to$ not reflexive

$(a, b) \in R$ but $(b, a) \notin R \to$ not symmetric

$(a, b) \in R & (b, c) \in R$ and also $(a, c) \in R \to$ transitive

(iii) R= ${(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a)}$

So, $(a, a), (b, b), (c, c) \in R \to$ Reflexive

$(a, b) \in R \to (b, a) \in R \to$ Symmetric

$(a, c) \in R \to (c, a) \in R$

$(b, a) \in R$ and $(a, c) \in R$

But $(b, c) \notin R \to$ not transitive

(iv) R= ${(a, a), (b, b), (c, c), (a, b), (b, c), (a, c)}$ is s relation in set A

So, $(a, a), (b, b), (c, c) \in R \to$ reflexive

$(a, b) & (b, c) \in R$ so, $(a, c) \in R \to$ transitive

$(a, b) \in R$ but $(b, a) \notin R \to$ not symmetric

(v) R= ${(a, a), (a, b), (b, a)}$

So, $(b, b) \notin R \to$ not reflexive

$(a, b) \in R$ and $(b, a) \in R \to$ symmetric

And $(a, b) \in R & (b, a) \in R$

and also $(a, a) \in R \to$ transitive

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R₁ = {(a, b) : a divide b}

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$= {\begin{array}{l} e^{l n x} = x & (0, 1) & e^{- x} \\ (- \infty, 0) & l n x & (1, \infty) \end{array}$

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

Which holds only for (0, 0)

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

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10. Given an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive.

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