Short Answer TypeGive 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.
R= {(L1, L2) : L1 is perpendicular to L2} Since no line can be perpendicular to itself ∴ R is not reflexive.
Let (L1, L2) ∈ R
∴ L1 is perpendicular to L2 ⇒ L2 is peipendicular to L1
⇒ (L2, L1) ∈ R
∴ (L1L2) ∈ R ⇒ (L2, L1) ∈ R
∴ R is symmetric
Again we know that if L1 is perpendicular to L2 and L2 is perpendicular to L3, then L1can never be perpendicular to L3.
∴ (L1, L2) ∈ R, (L2, L3) ∈ R does not imply (L1, L3) ∈ R
∴ R is not transitive.
Long Answer TypeDetermine 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}
Short Answer Type
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