Short Answer TypeLet 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.
A= {x ∈ Z : 0 ≤ x ≤ 12}
= {0, 1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12 } (i) R = {(a, b) : | a – b | is a multiple of 4|
As | a – a | = 0 is divisible by 4 ∴ (a, a) ∈ R ∀ a ∈ A.
∴ R is reflexive.
Next, let (a, b) ∈ R
⇒ | a – b | is divisible by 4
⇒ | – (b – a) | is divisible by 4 ⇒ | b – a | is divisible by 4 ⇒ (b, a) ∈ R
∴ R is symmetric.
Again. (a, b) ∈ R and (b, c) ∈ R
⇒ | a – A | is a multiple of 4 and | b – c | is a multiple of 4 ⇒ a – b is a multiple of 4 and b – c is a multiple of 4 ⇒ (a – b) + (b – c) is a multiple of 4 ⇒ a – c is a multiple of 4 ⇒ | a – c | is a multiple of 4 ⇒ (a, c) ∈ R ∴ R is transitive.
∴ R is an equivalence relation.
Set of elements which are related to | = {a ∈ A : (a, 1) ∈ R}
= {a ∈ A : |a – 1| is a multiple of 4]
= {1, 5, 9}
[ ∵ | 1 – 1 | = 0, | 5 – 1 | = 4 and I 9 – 1 | = 8 are multiples of 4] (ii) R = {(a, b) : a = b} ∴ a = a ∀ a ∈ A,
∴ R is reflexive.
Again, (a, b) ∈ R ⇒ a = b ⇒ b = a ⇒ (b,a) ∈ R ∴ R is symmetric.
Next. (a, b) ∈ R and (b, c) ∈ R
⇒ a = b and b = c
⇒ a = c ⇒ (a, c) ∈ R
∴ R is transitive.
∴ R is an equivalence relation.
Set of elements of A which are related to I = {a ∈ A : (a,1) ∈ R }
= {a ∈ A : a = 1} = {1}.
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