Let S be the subset of the set of ordered pairs of integers defined recursively byBasis step: (0,0) ? S.Recursive step: If (a,b) ? S, then (a, b + 1) ? S , (a + 1, b + 1) ? S, and (a + 2, b + 1) ? S.a) List the elements of S produced by the first four applications of the recursive definition.________________b) Use strong induction on the number of applications of the recursive step of the definition to show that a ? 2b whenever (a, b) ? S.________________c) Use structural induction to show that a ? 2b whenever (a, b) ? S.
1 | J a p a n e s e 5 0 Lecture 4 – April 11 th Main Events - Hogen Conflict (1156) - Heiji Conflict (1159) - Genpei Wars (1180-1185) Emergent Warrior Clans (Late Heian) - Minamoto 源 (Genji 源源) o Minamoto Yoritomo Sets up own government independent from Heian (even more powerful than Heian) Who made Minamoto successful - Taira 源 (Heike 源源) Kamakura Period Basics - “Kamakura” is a city in Eastern Japan - Location of the military government - Dates: 1185-1333 Main Events of Kamakura Period: - Yoritomo’s Death (1199) - Hojo Clan Rules Kamakura (1199-1333) - Jokyu Disturbance (1221): Emperor Go-Toba Rebellion o Wants to reassert power by invading the Hojo Clan o Total crushed by Hojo, Hojo gained more territories - Joei Code (1232): First Codification of Warrior Law Kamakura Period Power - System of Dual Government o Kyoto and Kamakura - Loss of power of cloistered Emperors (from 1221) - Kamakura control of provinces: o Stewards (Jito) and military governors (Shugo) - Mongol Invasions (1274,1281) 2 | J a p a n e s e 5 0 Outside of the Elite (Majority of Society) - Farming Peasants - Foragers, Hunters, Fisherman - Guilds (Craftsmen, Musicians, etc.) - Outcaste Groups War Tales - As Samurai gains power, war tales became more and more popular among the Japanese civilizatio - Hogen Conflict (1156) o Tale of Hogen - Heiji Conflict - Genpei Warfare Biwa Hoshi ( 琵琵琵琵) - Master of the Doctrines, who plays the Biwa - Plays for samurai, or deceased spirits Lecture 5 – April 13 th Review: Nara Buddhism - Major temples based in Nara capitol (urban) - Divisions based on study of six branches of doctrine 3 | J a p a n e s e 5 0 - Major Patrons: nobility, Fujiwara - Decline… Review: Heian Buddhism Tendai - Originates at Mt. Tiantai (Japanese=Tendai), Vhina - Japanese founder: Saicho - Center: Mt. Hiei (NE of Kyoto) - Main text: Lotus Sutra - Practices: Worship the Lotus Sutra, esoteric rituals Shingon (means mantra, literally “true word”) - Founder: Kukai - Center: Mt. Koya - Main text: Mahavairocana (Dainichi) Sutra - Concept: Awakening in this bodily existence - Practices: Esoteric rituals Divine Pantheon - Buddhas: o Fully awakened, liberated from cycle of rebirth o Reside in remote Buddha realms - Bodhisattvas 源源 : o Postpone becoming Buddha to save living beings o Reside in the world o Most popular: Bodhisattva Kannon 源源, Jizo 源源 - Kami (local deities): o Japanese local gods o Reside in the world, also seek Buddhist awakening Pure Land Buddhism - Founder: Honen - Mappo: Decline of the Dharma (state of the world) - Practive: Recitation the Nenbutsu (Name of Amida Buddha) - Goal: Deliverance to Amida’s pure land - Followers: all layers of society 4 | J a p a n e s e 5 0 Major Figures in Pure Land Buddhism - Genshin (942-1017) – Tendai priest. Essentials of Salvation. - Honen (1133-1212): Founder of Pure land school o Recitation of Nenbutsu – the only Method for salvation - Shinran (1173-1262): Founder of True Pure Land School o Humans must depend on “other power” of Amida o Faith in Amida - Ippen (1239-1288): Recitation of Amida’s name alone is enough o Faith is unnecessary Six Realms of Rebirth - Gods (Passions) - Humans (Optimal for Buddhist Path) - Asura (Anger, jealousy, war) - Beasts (Stupidity, servitude) - Hungry Ghosts (Starvation) - Hell (Torture) Week 3 Discussion: Rise of Samurais - Mongol Attacks on Japan (1274, 1281) Buddhism Doctrine - Four Noble Truths: o Life is suffering o Suffering has a cause o We can escape suffering 5 | J a p a n e s e 5 0 o To escape suffering, we must find a proper path - The Three Characteristics of Life: o Impermanence o Suffering o Illusion of self Tale of the Heike - Account of events leading to the Taira clan’s rise and their fall in the Genpei War (1180-1185) - Origins unknown - Most commonly known version is by Kakuishi, recorded in 1371