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A is the set of all polygons R = {(P 1 , P 2 ) : P 1 and P 2 have same number of sides } Since P and P have the same number of sides ∴ (P.P) ∈ R ∀ P ∈ A. ∴ R is reflexive. Let (P 1 , P 2 ) ∴ R ⇒ P 1 and P 2 have the same number of sides ⇒ P 2 and P 1 have the same number of sides ⇒ (P 2 , P 1 ) ∈ R ∴ (P 1 , P 2 ) ∈ R ⇒ (P 2 , P 1 ) ∈ R ∴ R is symmetric. Let (P 1 , P 2 ) ∈ R and (P 2 , P 3 ) ∈ R. ⇒ P 1 and P 2 have the same number of sides and P 2 and P 3 have same number of sides ⇒ P 1 and P 3 have the same number of sides ⇒ (P 1 , P 3 ) ∈ R ∴ (P 1 , P 2 ), (P 2 , P 3 ) ∈ R ∈ (P 1 , P 3 ) ∈ R ∴ R is transitive. ∴ R is an equivalence relation. Now T is a triangle. Let P be any element of A. Now P ∈ A is related to T iff P and T have the same number of sides P is a triangle required set is the set of all triangles in A.

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Relations and Functions

Short Answer Type

341. Determine whether each of the following relations are reflexive, symmetric and transitive :

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y) : x and y work at the same place}

(b) R = {(x,y) : x and y live in the same locality}

(d) R = {(x, y) : x is wife of y}

(e) R = {(x,y) : x is father of y}

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342. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

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343. Give an example of a relation which is
(i) Symmetric but neither reflexive nor transitive.

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344. Give an example of a relation which is
(ii) Transitive but neither reflexive nor symmetric.

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345. Give an example of a relation which is
(iii) Reflexive and symmetric but not transitive.

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346. Give an example of a relation which is
(iv) Reflexive and transitive but not symmetric.

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347. Give an example of a relation which is
(v) Symmetric and transitive but not reflexive.

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348.

Show that the relation R defined in the set A of all triangles as R = {(T₁, T₂) : T₁ is similar to T₂}, is equivalence relation. Consider three right angle triangles T₁ with Sides 3, 4, 5, T₂ with sides 5, 12, 13 and T₃ with sides 6, 8. 10. Which triangles among T₁, T₂ and T₃ are related ?

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349. Show that the relation R defined in the set A of all polygons as R = {(P₁, P₂) P₁ and P₂ have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle T with sides 3, 4 and 5 ?

A is the set of all polygons
R = {(P₁, P₂) : P₁ and P₂ have same number of sides }
Since P and P have the same number of sides
∴ (P.P) ∈ R ∀ P ∈ A.
∴ R is reflexive.
Let (P₁, P₂) ∴ R
⇒ P₁ and P₂ have the same number of sides ⇒ P₂ and P₁ have the same number of sides ⇒ (P₂, P₁) ∈ R
∴ (P₁, P₂) ∈ R ⇒ (P₂, P₁) ∈ R ∴ R is symmetric.
Let (P₁, P₂) ∈ R and (P₂, P₃) ∈ R.
⇒ P₁ and P₂ have the same number of sides and P₂ and P₃ have same number of sides
⇒ P₁ and P₃ have the same number of sides
⇒ (P₁, P₃) ∈ R
∴ (P₁, P₂), (P₂, P₃) ∈ R ∈ (P₁, P₃) ∈ R ∴ R is transitive.
∴ R is an equivalence relation.
Now T is a triangle.
Let P be any element of A.
Now P ∈ A is related to T iff P and T have the same number of sides P is a triangle required set is the set of all triangles in A.

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Long Answer Type

350. 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 = 2 x + 4.

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