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1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint. This requires two parts to be settled in the proof.

1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint. This requires two parts to be settled in the proof.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint. This requires two parts to be settled in the proof. 2. Let A Ø be a set. Prove that A? is an equivalence relation on A. 3. Prove that if a reflexive R (on some set A) satisfies x Ry A xRz yRz (1) for all x, y, z, then it is an equivalence relation. Caution. Note the order of the x, y, z in (1)! 4. Show that if for a relation R we know that R2 C R, then R is transitive, and conversely. Hint. There are two directions in the sought proof. 5. Without using the U R representation of R*, Prove that for any set relation R (we did not pick it to be "on" any particular set) R* = N{e:RCQA Q is transitive} Hint. You may use Questions 1 and 2 above even if you did not prove them. | Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not an equivalence relation. 6. %3D
Transcribed Image Text:1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint. This requires two parts to be settled in the proof. 2. Let A Ø be a set. Prove that A? is an equivalence relation on A. 3. Prove that if a reflexive R (on some set A) satisfies x Ry A xRz yRz (1) for all x, y, z, then it is an equivalence relation. Caution. Note the order of the x, y, z in (1)! 4. Show that if for a relation R we know that R2 C R, then R is transitive, and conversely. Hint. There are two directions in the sought proof. 5. Without using the U R representation of R*, Prove that for any set relation R (we did not pick it to be "on" any particular set) R* = N{e:RCQA Q is transitive} Hint. You may use Questions 1 and 2 above even if you did not prove them. | Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not an equivalence relation. 6. %3D
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