Prove that $A_n$ does not have a proper subgroup of index $<n$ for all $n\geq 5$.

Which of the two types of error is more serious? Explain.

Show that for every subgroup H of

$$S_n$$

for n ≥ 2, either all the permutations in H are even or exactly half of them are even.

A scientist working late at night in her low-temperature physics laboratory decides to have a cup of hot tea, but discovers the lab hot plate is broken. Not to be deterred, she puts about 8 oz of water, at

$$12 ^ { \circ } \mathrm { C }$$

from the tap into a lab dewar (essentially a large thermos bottle) and begins shaking it up and down. With each shake the water is thrown up and falls back down a distance of 33.3 cm. If she can complete 30 shakes per minute, how long will it take to heat the water to

$$87 ^ { \circ } \mathrm { C } ?$$

Would this really work? If not, why not?

The prime factorization of which number is

$$2 ^ { 5 } \times 5 ?$$

A. 50 B. 125 C. 160 D. 500

Let B(x) be "-10<x<10." Find the truth set of B(x) for each of the following domains. a. Z, b.

$$Z^+$$

, c. The set of all even integers

Let a > 0. Show that the volume obtained when the region between

$$y = a \sqrt { x - a x ^ { 2 } }$$

and the x-axis is rotated about the x-axis is independent of the constant a.

$$\text { Find a noncyclic subgroup of } A _ { 8 } \text { that has order } 4$$

Find the slope and y- intercept of the line, and draw its graph. x = -5

$$\text { Find a cyclic subgroup of } A _ { 8 } \text { that has order } 4$$

Describe two kernel data structures in which race conditions are possible. Be sure to include a description of how a race condition can occur.

State the center and radius of the circles described by the equations below. $x^{2}+y^{2}=289$

Determine whether the statement is true or false. If it is false, explain why.

$$a ^ { 2 } = b ^ { 2 } , \text { then } a = b$$

$$\left. \begin{array} { l } { \text { Suppose that } H \text { is a subgroup of } S _ { n } \text { of odd order. Prove that } H \text { is a } } \\ { \text { subgroup of } A _ { n } } \end{array} \right.$$