# Week 13 for Students.pdf - Relations Definition. A...

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RelationsDefinition. Arelation,𝑅, from a set𝐴to a set𝐵is a subset𝑅 ⊆ 𝐴 × 𝐵.For now, we will focus on relations on the same set, that is,𝐵 = 𝐴.1
Relations on a set and 3 possible propertiesDefinition. Arelation on a set𝑨is a subset𝑅 ⊆ 𝐴 × 𝐴. It is common toabbreviate the statement(𝑎, 𝑏) ∈ 𝑅as𝑎𝑅𝑏.SupposeRis a relation on a setA.1.RelationRisreflexiveif𝑥𝑅𝑥for every𝑥 ∈ 𝐴.i.e., if∀𝑥 ∈ 𝐴, 𝑥𝑅𝑥.(Examples.≤, ≥, =, |, ≡, ⊆; Not reflexive:<, >, ≠, ∤, ≢, ⊈)2.RelationRissymmetricif𝑥𝑅𝑦implies𝑦𝑅𝑥for all𝑥, 𝑦 ∈ 𝐴.i.e., if∀𝑥, 𝑦 ∈ 𝐴, 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥.(Examples.=, ≠, ≡, ≢; Not symmetric:<, ≤, >, ≥, |, ∤, ⊆)3.RelationRistransitiveif𝑥𝑅𝑦and𝑦𝑅𝑧implies𝑥𝑅𝑧for all𝑥, 𝑦, 𝑧 ∈ 𝐴.i.e., if∀𝑥, 𝑦, 𝑧 ∈ 𝐴, ((𝑥𝑅𝑦) ∧ (𝑦𝑅𝑧)) ⇒ 𝑥𝑅𝑧.(Examples.<, ≤, >, ≥, |, ∤, ⊆; Not transitive:𝑅 = {1,2 ,2,3 }.2
Graphical Ways to Describe Properties3
Here is a diagram for a relationon a setWrite the setsand. Determine ifis reflexive,symmetric, and/or transitive..4
Let𝐴 = {1,2,3,4,5,6}. Write out the relation𝑅that expresses|(divides)on𝐴. Then illustrate it with a diagram. Determine if𝑅is symmetric,reflexive, and/or transitive.5
Equivalence RelationsDefinition. A relationRon a set𝐴is anequivalence relationif it is reflexive,symmetric, and transitive.Examples:=, ≡, “has the same parity (odd/even)”, “has the same sign(+/-)”, “has

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