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Relations And Functions

Question
CBSEENMA12032246
Wired Faculty App

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 ?

Solution

R = {(T1, T2) : T1 is similar to T2}
Since every triangle is similar to itself
∴ R is reflexive.
Also (T1 T2) ∈ R ⇒ T1 is similar to T2 ⇒ T2 is similar to T1 ∴ (T2,T1) ⇒ R
∴ (T1,T2) ∈ R ⇒ (T2,T1) ∈ R ⇒ R is symmetric.
Again (T1, T2), (T2, T3) ∈ R
⇒ T1 is similar to T2 and T2 is similar to T3
∴ T1 is similar to T3 ⇒ (T1,T3) ∈ R ∴ (T1, T2), (T2,T3) ∈ R ⇒ (T1, T3) ∈ R ∴ R is transitive.
∴ R is reflexive, symmetric and transitive ∴ R is an equivalence relation.
Now T1, T2, T3 are triangles with sides 3, 4, 5 ; 5, 12, 13 and 6, 8, 10.
Since         3 over 6 equals 4 over 8 equals 5 over 10

∴ T1 is similar to T3 i.e. T3 is similar to T1.
No two other triangles are similar.

Some More Questions From Relations and Functions Chapter

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

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.

Give 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.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

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

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(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}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’. 

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