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Question
Let R be a relation defined on the set of natural numbers N as
R = {(x, y) : x, y ∈ N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.
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Solution
Domain of R is the values of x and range of R is the values of y that together should satisfy 2x+y = 41.
So,
Domain of R = {1, 2, 3, 4, ... , 20}
Range of R = {1, 3, 5, ... , 37, 39}
Reflexivity : Let x be an arbitrary element of R. Then,
x ∈ R
⇒ 2x+x = 41 cannot be true.
⇒ (x, x) ∉ R
So, R is not reflexive.
Symmetry :
Let (x,y) ∈ R.Then,
2x+y = 41
⇒ 2y+x = 41
⇒(y, x)∉R
So, R is not symmetric.
Transitivity :
Let (x, y) and (y, z)∈R
⇒ 2x+y = 41 and 2y + z = 41
⇒ 2x+z = 2x+ 41− 2y 41−y−2y = 41−3y
⇒ (x, z) ∉ R
Thus, R is not transitive.
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