Mark the tick against the correct answer in the following:
Let S be the set of all triangles in a plane and let R be a relation on S defined by ∆1 S ∆2⇔ ∆1 ≡ A2. Then, R is
According to the question ,
Given set S = {…All triangles in plane….}
And R = {(∆1 , ∆2) : ∆1 , ∆2∈ S and ∆1 ≡ ∆2}
Formula
For a relation R in set A
Reflexive
The relation is reflexive if (a , a) ∈ R for every a ∈ A
Symmetric
The relation is Symmetric if (a , b) ∈ R , then (b , a) ∈ R
Transitive
Relation is Transitive if (a , b) ∈ R & (b , c) ∈ R , then (a , c) ∈ R
Equivalence
If the relation is reflexive , symmetric and transitive , it is an equivalence relation.
Check for reflexive
Consider , (∆1, ∆1)
∴ We know every triangle is congruent to itself.
(∆1, ∆1) ∈ R all ∆1 ∈ S
Therefore , R is reflexive ……. (1)
Check for symmetric
(∆1 , ∆2) ∈ R then ∆1 is congruent to ∆2
(∆2 , ∆1) ∈ R then ∆2 is congruent to ∆1
Both the equation are the same and therefore will always be true.
Therefore , R is symmetric ……. (2)
Check for transitive
Let ∆1, ∆2, ∆3 ∈ S such that (∆1, ∆2) ∈ R and (∆2, ∆3) ∈ R
Then (∆1, ∆2)∈R and (∆2, ∆3)∈R
⇒∆1 is congruent to ∆2, and ∆2 is congruent to ∆3
⇒∆1 is congruent to ∆3
∴(∆1, ∆3) ∈ R
Therefore , R is transitive ……. (3)
Now , according to the equations (1) , (2) , (3)
Correct option will be (D)