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Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1 , T2 ) : T1 is congruent to T2 }. Show that R is an equivalence relation

RELATIONS AND FUNCTIONS Class 12, NCERT Chapter 1,  Example 2 Solution  R is reflexive since every triangle is congruent to itself. Further, (T1 , T2 ) ∈ R ⇒ T1 is congruent to T2 ⇒ T2 is congruent to T1 ⇒ (T2 , T1 ) ∈ R. Hence, R is symmetric. Moreover, (T1 , T2 ), (T2 , T3 ) ∈ R ⇒ T1 is congruent to T2 and T2 is congruent to T3 ⇒ T1 is congruent to T3 ⇒ (T1 , T3 ) ∈ R. Therefore, R is an equivalence relation. Example1     Example2     ←you are here Example3
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Example1 Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b): a is the sister of b} is the empty relation and R′ = {(a, b): the difference between heights of a and b is less than 3 meters} is the universal relation.

RELATIONS AND FUNCTIONS Class 12, NCERT Chapter1, Example1 Solution Since the school is a boy's school, no student at the school can be the sister of any student in the school. Hence, R = φ, showing that R is the empty relation. It is also obvious that the difference between the heights of any two students of the school has to be less than 3 meters. This shows that R′ = A × A is a universal relation.  Example1       ←you are here Example2 Example3
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