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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 }

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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 } 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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 } 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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 } 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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 } 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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 } 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= 𝑥1, 𝑥2 ∈ 𝑋𝑦𝑋 𝑥1 𝑠ℎ𝑧𝑛𝑓𝑡 𝑥𝑗𝑢ℎ 𝑥2 } 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:

Lesson 27 - Relations

 Partitions:

 Theorem: Let R be an equivalence relation on a set A, then

 Partition P = {[a] | a ∈ 𝐵} 𝑝𝑜 𝑡𝑓𝑢 𝐵

Lesson 27 - Relations

 Antisymmetric:

 𝑗𝑔 ∀𝑦 ≠ 𝑧 

𝑦, 𝑧 ∈ 𝑆 ⇒ 𝑧, 𝑦 ∉ 𝑆

Lesson 27 - Relations

 Antisymmetric:

 𝑗𝑔 ∀𝑦 ≠ 𝑧 

𝑦, 𝑧 ∈ 𝑆 ⇒ 𝑧, 𝑦 ∉ 𝑆

 Order Matters, consider set of to do list

Lesson 27 - Relations

 Antisymmetric:

 𝑗𝑔 ∀𝑦 ≠ 𝑧 

𝑦, 𝑧 ∈ 𝑆 ⇒ 𝑧, 𝑦 ∉ 𝑆

 Partial Order:

 R is reflexive  R is Antisymmetric  R is transitive

Homework (Group)

1.

Let A={0,1,2} and R={ (0,0),(0,1),(0,2),(1,1), (1,2), (2,2)} and S={(0,0),(1,1),(2,2)} be 2 relations on A. Show:

a)

R is a partial order relation.

b)

S is an equivalence relation.

2.

Give the partition p for x=y(mod4)

Homework (Individual)

1.

Which of these relations on { 0, 1, 2, 3} are equivalence relations? Give the types of relations.

a)

{ (0,0) , (1,1), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3) }

b)

{ (0,0) , (0,1), (0,2), (1,0), (1,1), (1,2), (2, 0), (2,2), (3,3) }

2.

Consider the relationship R where R = {(x,y) | x divides y}, is this a partial order relationship? Why

• r why not?