Short Answer TypeA is the set of all polygons
R = {(P1, P2) : P1 and P2 have same number of sides }
Since P and P have the same number of sides
∴ (P.P) ∈ R ∀ P ∈ A.
∴ R is reflexive.
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
∴ (P1, P2) ∈ R ⇒ (P2, P1) ∈ R ∴ R is symmetric.
Let (P1, P2) ∈ R and (P2, P3) ∈ R.
⇒ P1 and P2 have the same number of sides and P2 and P3 have same number of sides
⇒ P1 and P3 have the same number of sides
⇒ (P1, P3) ∈ R
∴ (P1, P2), (P2, P3) ∈ R ∈ (P1, P3) ∈ R ∴ R is transitive.
∴ R is an equivalence relation.
Now T is a triangle.
Let P be any element of A.
Now P ∈ A is related to T iff P and T have the same number of sides P is a triangle
required set is the set of all triangles in A.
Let R be the relation defined on the set of natural numbers N as R = {(x, y) : x ∈ N, y ∈ N, 2 x + y = 41 }
Find the domain and range of this relation R. Also verify whether R is (i) reflexive (ii) symmetric (iii) transitive.
The following three relations are defined on the set of natural numbers :
R = {(x, y) : x < y, x ∈ N, y ∈ N}
S = { (x,y) : x + y = 10, x ∈ N, y ∈ N}
T = { (x, y) : x = y or x – y = 1, x ∈ N, y ∈ N } Explain clearly which of the above relations are (i) Reflexive (ii) Symmetric (iii) Transitive.
Long Answer TypeShow that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12 }, given by
(i) R = {(a, b) : | a – b | is a multiple of 4 }
(ii) R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.
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