CSCI 246 Class 16 MORE EQUIVALENCE RELATIONS Quiz Questions - PowerPoint PPT Presentation

CSCI 246 – Class 16 MORE EQUIVALENCE RELATIONS

Quiz Questions  Lecture 27:  What is the difference between an equivalence relation and a partial order relation?

Notes and Clarifications  Extra credit due tonight  Quiz option for tomorrow  Let’s talk relations

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜}

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜} examples in D include: (3.4, 6.4), (9.11, -12.11)

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜} examples in D include: (3.4, 6.4), (9.11, -12.11)  what about: (1/3, 4/3)? (5,8)

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜} examples in D include: (3.4, 6.4), (9.11, -12.11)  what about: (1/3, 4/3)? (5,8)  Is this a reflexive relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜} examples in D include: (3.4, 6.4), (9.11, -12.11)  what about: (1/3, 4/3)? (5,8)  Is this a symmetric relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜} examples in D include: (3.4, 6.4), (9.11, -12.11)  what about: (1/3, 4/3)? (5,8)  Is this a Transitive relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation D, 𝐸 = 𝑦, 𝑧 ∈ ℝ𝑦ℝ 𝑦, 𝑧 ℎ𝑏𝑤𝑓 𝑢ℎ𝑓 𝑡𝑏𝑛𝑓 𝑒𝑓𝑑𝑗𝑛𝑏𝑚 𝑓𝑦𝑞𝑏𝑜𝑡𝑗𝑝𝑜} examples in D include: (3.4, 6.4), (9.11, -12.11)  what about: (1/3, 4/3)? (5,8)  Is this an equivalence relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 }

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 } Example ordered pairs (cat, hat), (book, nook)

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 } Example ordered pairs (cat, hat), (book, nook)t Is (Bobcat, Bobcat) in R? Is (backpack, knack) in R?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 } Example ordered pairs (cat, hat), (book, nook)t Is (Bobcat, Bobcat) in R? Is (backpack, knack) in R? Is this a reflexive relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 } Example ordered pairs (cat, hat), (book, nook)t Is (Bobcat, Bobcat) in R? Is (backpack, knack) in R? Is this a symmetric relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 } Example ordered pairs (cat, hat), (book, nook)t Is (Bobcat, Bobcat) in R? Is (backpack, knack) in R? Is this a transitive relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Consider the relation R, R = 𝑥 1 , 𝑥 2 ∈ 𝑋𝑦𝑋 𝑥 1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥 2 } Example ordered pairs (cat, hat), (book, nook)t Is (Bobcat, Bobcat) in R? Is (backpack, knack) in R? Is this an equivalence relationship?

Lesson 25 and 26 - Relations  Remember: Relation Types: Let R be a relation on a set A  Reflexive: ∀𝑏 ∈ 𝐵, 𝑏, 𝑏 ∈ 𝑆  Symmetric: ∀𝑏, 𝑏 ′ ∈ 𝐵, 𝑏, 𝑏 ′ ∈ 𝑆 ⟹ 𝑏 ′ , 𝑏 ∈ 𝑆  Transitive: ∀𝑏, 𝑏 ′ , 𝑏 ′′ , 𝑏, 𝑏 ′ , 𝑏 ′ , 𝑏 ′′ ∈ 𝑆 ⟹ 𝑏, 𝑏′ ∈ 𝑆 Revisiting yesterday’s homework: