Short Answer TypeL is the set of all lines in XY plane.
R = {(L1, L2) : L1 is parallel to L2}
Since every line l ∈ L is parallel to itself,
∴ (l,l) ∴ R ∀ l ∈ L
∴ R is reflexive.
Let (L1, L2) ∈ R ∴ L1 || L2 ⇒ L2 || L1
⇒ (L2, L1) ∈ R.
∴ R is symmetric.
Next, let (L1 L2) ∈ R and (L2, L3) ∈ R ∴ L1 || L2 and L2 || L3
∴ L1 || L3 (L1 , L3) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
Let P be the set of all lines related to the line y = 2 x + 4.
∴ P = {l : l is a line related to the line y = 2 x + 4}
= {l : l is a line parallel to the line y = 2 x + 4}
= { l : l is a line with equation y = 2 x + c, where c is an arbitrary constant }
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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