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Relations and Functions
A 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.
Sponsor Area
Give an example of a relation which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.
Determine whether each of the following relations are reflexive, symmetric and transitive :
(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as
R = {(x, y) : 3 x – y = 0}
(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}
(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}
Sponsor Area
Sponsor Area