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For any binary relation ▷ on a set X, define the binary relation by letting xy we do not have ▷y. Let be a binary relation on a set X, and define binary relations ~ letting I~y ⇒ï≥y and yx, and and by I>Y ⇒ Iy and yr. (a) Argue that, if relation is complete and transitive, then the relation~~ is reflexive (rr), transitive (~y and y~ z implies that ~ z), and symmetric (ry implies that yr). [Said differently, you're showing is an equivalence relation.] 2 (b)Argue that, if relation is complete and transitive, then the relation is asymmetric (ry implies that yr) and negative transitive (ry and y z implies that rz).

For any binary relation ▷ on a set X, define the binary relation by letting xy we do not have ▷y. Let be a binary relation on a set X, and define binary relations ~ letting I~y ⇒ï≥y and yx, and and by I>Y ⇒ Iy and yr. (a) Argue that, if relation is complete and transitive, then the relation~~ is reflexive (rr), transitive (~y and y~ z implies that ~ z), and symmetric (ry implies that yr). [Said differently, you're showing is an equivalence relation.] 2 (b)Argue that, if relation is complete and transitive, then the relation is asymmetric (ry implies that yr) and negative transitive (ry and y z implies that rz).

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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2. For any binary relation ▷ on a set X, define the binary relation by letting xy ⇒ we do not have Dy. Let be a binary relation on a set X, and define binary relations and by letting I~Y Iy and yr, and I>y⇒ I≥ y and y Z x. (a) Argue that, if relation is complete and transitive, then the relation is reflexive (rr), transitive (r~y and y~ z implies that ~ z), and symmetric (ry implies that yr). [Said differently, you're showing ~ is an equivalence relation.] (b) Argue that, if relation is complete and transitive, then the relation > is asymmetric (ry implies that yr) and negative transitive (ry and y z implies that xz).
Transcribed Image Text:2. For any binary relation ▷ on a set X, define the binary relation by letting xy ⇒ we do not have Dy. Let be a binary relation on a set X, and define binary relations and by letting I~Y Iy and yr, and I>y⇒ I≥ y and y Z x. (a) Argue that, if relation is complete and transitive, then the relation is reflexive (rr), transitive (r~y and y~ z implies that ~ z), and symmetric (ry implies that yr). [Said differently, you're showing ~ is an equivalence relation.] (b) Argue that, if relation is complete and transitive, then the relation > is asymmetric (ry implies that yr) and negative transitive (ry and y z implies that xz).
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