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Please write an essay in response to one of the two essay questions below.(50 points) From 600 to 1200 C.E. Islam spread from Arabia eastward and westward to northern Africa, Spain, India, Asia, and central and eastern Europe.  Identify and discuss the factors in this expansion, and explain the role each factor played in this expansion.  Cite examples to develop and substantiate your points, and cover the major events and key groups and individuals in the rise of Islam during this period. Between 600 and 1200 a resurgent China dominated eastern Asia.  Explain how the Sui, the Tang, and the Song Dynasties of China dominated eastern Asia during this period.  In your essay identify and discuss examples of Chinese dominance of east Asia, factors that explain this dominance and the major events which mark it.  In addition, discuss the responses of Japan, Korea, and southeastern Asia to Chinese domination, as well as the key events in the histories of those countries.

Which of the following approaches do we use  to prove that  A ⊆ B ? {“version”:”1.1″,”math”:”\text{Which of the following approaches do we use }\\ \text{to prove that } A\subseteq B ?”}

The coefficient of x19y6 in the expansion of (x+y2)25 is {“version”:”1.1″,”math”:”\text{The coefficient of }x^{19}y^6 \text{ in the expansion of } (x+y^2)^{25} \text{ is } “}

Select all anti-symmetric relations.

Suppose a box contains 12 apples, 15 oranges and 20 pears. How many fruits do you have to pick from the box to guarantee that you have at least 12 oranges?

If a relation R on a set is a total ordering, then R must also be a partial ordering.

When proving the statement,  `if  x 3  is irrational then  x  is irrational’   using contradiction we assume that  x 3  is rational. {“version”:”1.1″,”math”:”\text{When proving the statement, } \\ \textbf{`if $x^3$ is irrational then $x$ is irrational’ }\\ \text{ using contradiction we assume that } x^3 \text{ is rational.}”} 

Consider the relation R defined on S, where S is the set of all cars, and R={(x,y) | x and y are red cars}. Select all true statements.

Consider the relation R on S, where S is the set of all people, and R={(G,H) | G and H have the same number of biological children}.  Select all true statements.

Suppose a group consists of five males and seven females. How many ways can you select a committee of four members from the group, if the committee is required to have at least two females?