Question

Exercise 2. Give an example of a relation on a set that is both symmetric and transitive but not reflexive. Explain what is wrong with the following "proof." Statement: If R is symmetric and transitive, then R is reflexive. "Proof": Suppose R is symmetric and transitive. Symmetric means that x R y implies y R x. We apply transitivity to x R y and y R x to give x R x. Therefore R is reflexive.

          Exercise 2. Give an example of a relation on a set that is both symmetric and transitive but not reflexive.
Explain what is wrong with the following "proof."
Statement: If R is symmetric and transitive, then R is reflexive.
"Proof": Suppose R is symmetric and transitive. Symmetric means that x R y implies y R x. We apply transitivity to x R y and y R x to give x R x. Therefore R is reflexive.

Exercise 2. Give an example of a relation on a set that is both symmetric and transitive but not reflexive.
Explain what is wrong with the following "proof."
Statement: If R is symmetric and transitive, then R is reflexive.
"Proof": Suppose R is symmetric and transitive. Symmetric means that x R y implies y R x. We apply transitivity to x R y and y R x to give x R x. Therefore R is reflexive.

Added by Brian M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Keondre Parker

11/01/2022

Step 1

This relation is symmetric because if (x, y) is in R, then (y, x) is also in R. It is also transitive because if (x, y) and (y, z) are in R, then (x, z) is also in R. However, this relation is not reflexive because there is no pair (c, c) or (d, d) in R. Show more…

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Give an example of a relation on a set that is both symmetric and transitive but not reflexive. Explain what is wrong with the following "proof." Statement: If R is symmetric and transitive, then R is reflexive. "Proof": Suppose R is symmetric and transitive. Symmetric means that x R y implies y R x. We apply transitivity to x R y and y R x to give x R x. Therefore R is reflexive.