Let and be sets. A binary relation from to is a subset of . If - PowerPoint PPT Presentation

�✁ ❦ ❥ ◗ ❯❵ ❚ ❙ ◗ P ❴ ❳ ❧ ❲ ▲ ❜ ❥ ❛ ❚ ❳ ◗ ❯❵ ❚ ❙ ◗ P ❴ ❳ ✐ ❲ ▲ ❜ ❚ ♠ ❳ ❅ ✳ ❅ ✻ ✾ ✳ ❅ ❉ ❈ ✵ ❇ ❂ ❃❆ ✹ ✷ ❄ ❜ ❂❃ ❁ ❀ ✿ ✸ ✽✾ ✼ ✻ ✺ ✸✹ ✷ ✶ ✵ ✲✳✴ ❚ ◗ ❋ ❙ ❙ ◗ P ❴ ❳ ❝ ❲ ▲ ❜ ❚ ❛ ◗ ❯❵ ✂✄ ◗ ❯❵ P ❴ ❳ ❫ ❲ ▲ ✂ ✆ ❍ ❪ ❭ ■ ❩❬ ✾ ❚ ◗ ❯❵ ❢ ❚ ❙ ◗ P ❴ ❳ ❤ ❲ ▲ ❜ ❚ ❣ ❳ ◗ ❚ ❞ ❳ ◗ ❯❵ ❚ ❙ ◗ P ❴ ❳ ❡ ❲ ▲ ❜ ❚ ❊ ✻ ❋ ❲ ① ① ❙ ♠ ❙ ✇ ❙ ❦ ❳ ✈ ❲ ④ ▼ ▲ ✲✳✴ ❲ s ❳ ❫ ③ ❲ ❳ ❫ ❲ ① ① ① ❙ ♠ ① ✵ ✇ ❃❆ ✻ ❋ ❊ ✳ ❅ ✻ ✾ ✳ ❅ ❉ ❈ ✵ ❇ ❂ ✹ ✶ ✷ ❅ ❄ ❂❃ ❁ ❀ ✿ ✸ ✽✾ ✼ ✻ ✺ ✸✹ ✷ ❙ ❙ ✽ ❏ ❙ ◗ P q ❲ r ◆ q ◆ ▼ ❲ ▲ ✂ ❑ ✁ ❯ ✝ ■ ❍ ✆ ● ♣ ❑ ✁ ❑ ✁ ♦ ❭ ❏ ♥ ❨ ◗ ❦ ❙ ❳ ✈ ▼ ❲ ▲ ❲ s q q ❲ q ❱ ❯ ❨ ❚ ❱ P ❲ ❚ ❙ ◗ P ◆ ❱ ❚ r ❱ ❨ ❙ ▼ ✻ ❚ ❊ ✻ ✁ ✝ ■ ❍ ✆ ● ❋ ✻ ❋ ❊ ✳ ❅ ✾ ❑ ✳ ❅ ❉ ❈ ✵ ❇ ❂ ❃❆ ✹ ✷ ❅ ❄ ❏ ✂ ❁ ❚ ◗ ❙ ◗ P ▼ ❲ ▲ ▼ ▼ ▼ ▲ ❲ ◗ ▲ ✳ ❲ ◗ ❲ ❚ ❙ ◆ ❖ ▼ ◆ ▼ ◆ ▼ ❂❃ ❀ ◗ ✡ ✘ ✖✗ ✕ ✔ ✓ ✒ ✑ ✏ ✎ ✍ ☞✌ ☛ ✠ ✚ ✂ ✆ ☎ ✟ ✝ ✄ ✟ ☎ ✝ ✞ ✆ ✆✝ ☎ ✙ ✛ ✿ ✬✮ ✸ ✽✾ ✼ ✻ ✺ ✸✹ ✷ ✶ ✵ ✲✳✴ ✰✱ ✯ ✫✬✭ ✗ ✪ ✩ ✦ ✙ ★ ✦✧ ✙ ✥ ✤ ✣ ✢ ✖ ✜ ❲ ❚ ❱ ▼ ❨ ❙ ◗ P ❲ ❨ ❙ ❚ P ❲ ❚ ❙ ◗ P ❲ ◗ ◗ ❲ P ❚ ❙ ◗ ❲ ▼ ▲ ❚ ◗ ❙ ❚ ❱ ▼ ❲ ❙ ❙ ❄ ❅ ✻ ✾ ✳ ❅ ❉ ❈ ✵ ❇ ❂ ❃❆ ✹ ✷ ❅ ❂❃ ❚ ✸✹ ❱ ▼ ✲✳✴ ✵ ✶ ✷ ✺ ❁ ✻ ✼ ✽✾ ✸ ✿ ❀ ❚ ❳ ▼ P ❚ ❙ ◗ ❲ ❚ ❙ ◗ ❲ ▼ ◗ ❙ ❚ P ▼ ❲ ▲ ❱ ✼ ▲ ❲ Let and be sets. A binary relation from to is a subset of . If , we write and say is related to by . P❘◗ ❯❘❱ A relation on the set is a relation from to . A relation on a set is called reflexive if for every element . ❯❘❱ A relation on a set is called symmetric if whenever ❯❘❱ , for . ❯❘❱ A relation on a set such that and only if , P❘◗ ❯❘❱ ❯❘❱ for , is called antisymmetric . A relation on a set is called transitive if whenever and , ❯❘❱ ❯❘❱ then , for . ❯❘❱ Let be a relation from a set to a set and be a relation from to a set . The composite of and is the relation consisting of ordered pairs , where , and for which there exists an element such that and . ❯❘❱ We denote the composite of and by . Let be a relation on the set . The powers , , are ❲✉t defined inductively by and . ❲②t ❲②t Theorem : The relation on a set is transitive if and only if for . ❲✉t

♥ ✸ ✹ ✷ ❅ ❄ ❂❃ ❁ ❀ ✿ ✽✾ ❂ ✼ ✻ ✺ ✸✹ ✷ ✶ ✵ ✲✳✴ ❽ ❃❆ ❇ ❚ ✻ ✁ ✝ ✄ ✆ ❑ ❑ ❏ ♥ ❊ ❋ ✵ ❊ ✳ ❅ ✻ ✾ ✳ ❅ ❉ ❈ ❯ ❙ ✁ ❽ ❯ ❝ ❺ ❙ ❫ ❍ ❙ ❲ ❯ ❲ ❫ ❺ ❙ ◗ P ❚ ❙ ❫ ❼ ❽ ❙ ◗ ❯ P ❚ ◗ ✈ ▼ ❲ ▲ ❲ ❽ ❚ ① ❙ ❫ ❼ t ❺ P ❙ ① ① ❾ ✝ ❺ ✲✳✴ ✽✾ ✼ ✻ ✺ ✸✹ ✷ ✶ ✵ ▲ ✿ ▲ ▲ ✂ ❑ ❏ ✁ ✝ ✁ ✝ ✸ ❀ ■ ❉ ✻ ❋ ❊ ✳ ❅ ✻ ✾ ✳ ❅ ❈ ❁ ✵ ❇ ❂ ❃❆ ✹ ✷ ❅ ❄ ❂❃ ☎ ➂ ❿ ❲ ✽✾ ✼ ✻ ✺ ✸✹ ✷ ✶ ✵ ✲✳✴ ▲ ✿ ❲ ❚ ◗ ❯ ❚ ❙ ▼ ❲ ▲ ✸ ❀ ❅ ❉ ✻ ❋ ❊ ✳ ❅ ✻ ✾ ✳ ❅ ❈ ❁ ✵ ❇ ❂ ❃❆ ✹ ✷ ❅ ❄ ❂❃ t ➃ ① ● ❸ ❀ ❷ ▲ ✂ ■ ☎ ❷ ✆ ⑨ ① ❍ ■ ✝ ✁ ✿ ✸ ❑ ✸✹ ❁ ❫ ❺ ❙ ❻ ✷ ✺ q ❹ ✻ ❚ ✼ ◗ ✽✾ ▲ ⑧ ■ P ❈ ❅ ✻ ✾ ✳ ❅ ❉ ✵ ❊ ❇ ❂ ❃❆ ✹ ✷ ❅ ❄ ✳ ❋ ❏ ✁ ✂ ❑ ❑ ✂ ❏ ▲ ✝ ✻ ⑥ q ■ ❍ ✆ ● ⑦ ❙ ❯ ❺ ❙ t ❺ ❙ ① ① ① ❫ ✈ ❺ ❙ ❻ ❺ q ❫ ⑥ ❲ ⑥ ❏ ❺ ① ❙ ❝ ❺ ❙ ❫ ❙ ▲ ◗ ❲ ❲ ⑥ ❚ ◗ q ❳ ❚ t t ❏ ⑤ t ❺ ❙ ❫ ❼ ✶ ❯ ① ① ① ❙ ❯ ❝ ❺ ❙ ✂ ❂❃ ✵ ◗ ❺ ✟ ✲✳✴ ❺ ❲ ☎ ❻ ❹ ✂ ❳ ✆ ❪❶⑩ Let be a property of relations (transitivity, refexivity, symmetry). A relation A directed graph , or digraph , consists of a set of vertices (or nodes) is losure of w.r.t. if and only if has property , contains , together with a set of ordered pairs of elements of called edges and is a subset of every relation with property containing . (or arcs). A path from to in the directed graph is a sequence of one or more edges in , where P❘❺ P❘❺ and . This path is denoted by and has length . A path that begins and ends at the same vertex is called a circuit or cycle. There is a path from to in a relation is there is a sequence of elements with . P❘❺ Theorem: Let be a relation on a set . There is a path of length from to if and only if . ❲✉t Let be a relation on a set . The connectivity relation consists of We want to use relations to form partitions of a group of students. Each ❲➁➀ pairs such that there is a path between and in . member of a subgroup is related to all other members of the subgroup, P❘◗ but to none of the members of the other subgroups. Theorem: The transitive closure of a relation equals the connectivity relation . Use the following relations: ❲➁➀ Partition by the relation ”older than” Partition by the relation ”partners on some project with” Partition by the relation ”comes from same hometown as” Which of the groups will succeed in forming a partition? Why?

❩ ➊ ➌ ❙ ➍ ❽ ➎ ➎ q q ▼ ➌ ➉ ❳ ➍ q ❽ ➎ ▼ ➌ ➈ ▼ ➏ ❳ ➊ ➍ ➉ ❳ ➐ ▼ ▲ ➌ ❏ ■ ❍ ✆ ❑ ✄ ✆ ● ➄ ❍ ■ ✝ ✁ ❑ ✂ ✂ ■ ❑ ⑧ ➂ ■ ☎ ✝ ✁ ✝ ✁ ❏ ❑ ➑ ➒ ✁ ❃❆ ✻ ✼ ✽✾ ✸ ✿ ❀ ❁ ❂❃ ❄ ❅ ✷ ✹ ❂ ✸✹ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✻ ❅ ✳ ❊ ❋ ✻ ✺ ✷ ➓ ❵ ▼ ➌ ❳ q ▲ ❲ q ❲ q ❴ ▼ ➌ ➍ ✶ ❽ ➎ ❜ q ❲ ▼ ➌ ❙ ➍ ❽ ➎ ✲✳✴ ✵ ❾ ✆ ✟ ➇ ◗ ➆ ➇ ◗ ❲ ❚ ❽ ➅ ◗ ➆ ❚ ◗ ▲ ➄ ▼ ◗ ❲ ❚ ➅ ◗ ➆ ❳ ➅ ➅ ▼ ➆ ✆ ✟ ✁ ❾ ■ ❍ ✆ ❑ ✄ ✆ ● ❍ ◗ ■ ✝ ✁ ❏ ❑ ✂ ▲ ▼ ▲ ❲ ▼ ❚ ❲ ➔ ❁ ✸✹ ❅ ✺ ✻ ❋ ❊ ✼ ✽✾ ✸ ✿ ✳ ❀ ❂❃ ✶ ❄ ❅ ✷ ✹ ❃❆ ❂ ❇ ✵ ❈ ❉ ❅ ✳ ✾ ✷ ✻ ➊ ❩ ❳ ➋ ➉ ❚ ✲✳✴ ✵ ➅ ➈ ✻ ➆ ◗ ➅ ➆ A relation on a set is called an equivalence relation if it is reflexive, A partition of a set is a collection of disjoint nonempty subsets symmetric, and transitive. Two elements that are related by an (where is an index set) of that have as their union: equivalence relation are called equivalent. for , when Let be an equivalence relation on a set . The set of all elements that are related to an element of is called the equivalence class of . : equivalence class of w.r.t. . If then is representative of this equivalence class. Theorem: Let be an equivalence relation on a set . Then the equivalence classes of form a partition of . Conversely, given a Theorem: Let be an equivalence relation on a set . The partition of the set , there is an equivalence relation that following statements are equivalent: has the sets , as its equivalence classes. (1) (2) (3)