. Let A = {1,2,3,4}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule     (X,Y) ∈ R ⇔ (X ∩ Y = Ø) ∧ (X ∪ Y = A). (a) Prove or find a counterexample for the following properties and based on the findings from the properties, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.: antisymmetry asymmetry symmetry irreflexivity reflexivity transitivity uniqueness

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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1. Let A = {1,2,3,4}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule

    (X,Y) ∈ R ⇔ (X ∩ Y = Ø) ∧ (X ∪ Y = A).

(a) Prove or find a counterexample for the following properties and based on the findings from the properties, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.:

  • antisymmetry
  • asymmetry
  • symmetry
  • irreflexivity
  • reflexivity
  • transitivity
  • uniqueness
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