Set, Relation and FunctionEasy
Question
Let R be the real line. Consider the following subsets of the plane R × R.
S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x - y is an integer}. Which one of the following is true?
S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x - y is an integer}. Which one of the following is true?
Options
A.neither S nor T is an equivalence relation on R
B.both S and T are equivalence relations on R
C.S is an equivalence relation on R but T is not
D.T is an equivalence relation on R but S is not
Solution
T = {(x, y) : x-y ∈ I}
as 0 ∈ I T is a reflexive relation.
If x - y ∈ I ⇒ y - x ∈ I
∴ T is symmetrical also
If x - y = I1 and y - z = I2
Then x - z = (x - y) + (y - z) = I1 + I2 ∈ I
∴ T is also transitive.
Hence T is an equivalence relation.
Clearly x ≠ x + 1 ⇒ (x, x) ∉ S
∴ S is not reflexive.
as 0 ∈ I T is a reflexive relation.
If x - y ∈ I ⇒ y - x ∈ I
∴ T is symmetrical also
If x - y = I1 and y - z = I2
Then x - z = (x - y) + (y - z) = I1 + I2 ∈ I
∴ T is also transitive.
Hence T is an equivalence relation.
Clearly x ≠ x + 1 ⇒ (x, x) ∉ S
∴ S is not reflexive.
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