Short Answer Type(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}
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 ?
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.
∴ T1 is similar to T3 i.e. T3 is similar to T1.
No two other triangles are similar.
Long Answer Type
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