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Show that the Relation R on ℝ Defined as R = {(A, B): a ≤ B}, is Reflexive, and Transitive but Not Symmetric. - Mathematics

Sum

Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.

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Solution

R = {(a, b); a ≤ b}
Clearly (a, a) ∈ R as a = a.
R is reflexive.
Now,
(2, 4) ∈ R (as 2 < 4)
But, (4, 2) ∉ R as 4 is greater than 2.
∴ R is not symmetric.
Now,
let (a, b), (b, c) ∈ R.
Then,
a ≤ b and b ≤ c
⇒ a ≤ c
⇒ (a, c) ∈ R
 R is transitive.
Hence, R is reflexive and transitive but not symmetric.
  Is there an error in this question or solution?
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