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In 1260, Muslim Mamluk soldiers defeated Mongol forces during [BLANK-1]. This marked the first time that a major Mongol invasion had been repelled. The victory also likely saved the majority of Muslim lands in Africa and Europe from falling into the hands of the Mongols.

Peter Abelard was the most famous professor at the University of Paris and his writing exemplified the intellectual movement known as Scholasticism. In his book, [BLANK-1], a classic of Scholasticism which literally means “Yes and No,” he discussed inconsistencies in the Bible. With this book, Abelard examined the inconsistencies by writing in defense of both sides of each question.

Berbers came to dominate [BLANK-1], an important overland commercial network.

[BLANK-1] were widely used by monks and others during the Crises of the Ninth and Tenth Centuries in Europe. They were considered the “weapons of the weak,” which consisted mainly of words and spiritual belief, and were used against invading Vikings, Magyars, and other raiders when they attacked the weakly defended towns and monasteries of Europe. The monks would call upon God to harm their enemies and to protect them in an increasingly violent world.

A period of dynastic hereditary rule in Persia known as [BLANK-1] lasted from 224-651 C.E. This Persian dynasty was founded by Ardashir I and this society became the chief enemy and rival of the Byzantine Empire. Ultimately, long-term wars with the Byzantines weakened this civilization and left them ripe for conquest by Muslim armies in the seventh century.

An eight-century issue known as [BLANK-1] was a serious crisis faced by the Byzantine Empire. This theological dispute raised questions about the power of the emperor to intervene in religious disputes. It also led the Orthodox Church and Roman Catholic Church to sever ties – a division that would become permanent in 1054 with the East-West Schism.

Read the following abstract from a journal article and identify the level of measurement for the independent variable: “We examine the relative pay of occupations involving care, such as teaching, counseling, providing health services, or supervising children. We use panel data from the National Longitudinal Survey of Youth covering workers between 17 and 35 years of age. Care work pays less than other occupations after controlling for the education and employment experience of the workers, many occupation and industry characteristics, and … unmeasured, stable characteristics of those who hold the jobs.” Paula England, Michelle Budig, Nancy Folbre, Wages of Virtue: The Relative Pay of Care Work, Social Problems, Volume 49, Issue 4, 1 November 2002, Pages 455–473,

Work out the following problem by hand and prove the following. Upload a photo of your work. \( \exists x [(Kx \land \forall y (Ky \rightarrow x = y)) \land Bx], \quad Ka \quad \therefore \quad \exists x (Bx \land x = a) \)

Work out the following problem by hand and show that each of the following arguments is valid using a natural deduction proof in FOL. Upload a photo of your work. \( \forall x (Lx \rightarrow Gx), \quad \exists x Lx \quad \therefore \quad \exists x Gx \)

Use the Horizontal Line Test to determine whether the following statement is true or false. ​ The function  is one-to-one on its entire domain and therefore has an inverse function. ​