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Question
Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?
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Solution
We observe the following properties on R:
Reflexivit:
Let P1 be an arbitrary element of A.
Then, polygon P1 and P1 have the same number of sides, since they are one and the same.
⇒ (P1, P1) ∈ R for all P1 ∈ A
So, R is reflexive on A.
Symmetry:
Let (P1, P2) ∈ R
⇒ P1 and P2 have the same number of sides.
⇒ P2 and P1 have the same number of sides.
⇒ (P2, P1) ∈ R for all P1, P2 ∈ A
So, R is symmetric on A.
Transitivity:
Let (P1, P2), (P2, P3) ∈ R
⇒ P1 and P2 have the same number of sides and P2 and P3 have the same number of sides.
⇒ P1, P2 and P3 have the same number of sides.
⇒ P1 and P3 have the same number of sides.
⇒ (P1, P3) ∈ R for all P1, P3 ∈ A
So, R is transitive on A.
Hence, R is an equivalence relation on the set A.
Also, the set of all the triangles ∈ A is related to the right angle triangle T with the sides 3, 4, and 5.
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