Help me with appropriate answer..

L is the set of all lines in XY plane. R = {(L 1 , L 2 ) : L 1 is parallel to L 2 } Since every line l ∈ L is parallel to itself, ∴ (l,l) ∴ R ∀ l ∈ L ∴ R is reflexive. Let (L 1 , L 2 ) ∈ R ∴ L 1 || L 2 ⇒ L 2 || L 1 ⇒ (L 2 , L 1 ) ∈ R. ∴ R is symmetric. Next, let (L 1 L 2 ) ∈ R and (L 2 , L 3 ) ∈ R ∴ L 1 || L 2 and L 2 || L 3 ∴ L 1 || L 3 (L 1 , L 3 ) ∈ R ∴ R is transitive. Hence, R is an equivalence relation. Let P be the set of all lines related to the line y = 2 x + 4. ∴ P = {l : l is a line related to the line y = 2 x + 4} = {l : l is a line parallel to the line y = 2 x + 4} = { l : l is a line with equation y = 2 x + c, where c is an arbitrary constant }

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

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}

166 Views

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.

139 Views

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

151 Views

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

156 Views

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

139 Views

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

135 Views

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

171 Views

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 ?

150 Views

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 ?

172 Views

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.

L is the set of all lines in XY plane.
R = {(L₁, L₂) : L₁ is parallel to L₂}
Since every line l ∈ L is parallel to itself,
∴ (l,l) ∴ R ∀ l ∈ L
∴ R is reflexive.
Let (L₁, L₂) ∈ R ∴ L₁ || L₂ ⇒ L₂|| L₁⇒ (L₂, L₁) ∈ R.
∴ R is symmetric.
Next, let (L₁ L₂) ∈ R and (L₂, L₃) ∈ R ∴ L₁ || L₂ and L₂|| L₃∴ L₁|| L₃ (L₁, L₃) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
Let P be the set of all lines related to the line y = 2 x + 4.
∴ P = {l : l is a line related to the line y = 2 x + 4}
= {l : l is a line parallel to the line y = 2 x + 4}
= { l : l is a line with equation y = 2 x + c, where c is an arbitrary constant }