A Boolean algebra may also be defined as a partially ordered set with certain additional properties. Let $(B, \leqslant)$ be a partially ordered set. For any $x, y \in B,$ we define the least upper bound of x and y as an element z such that $x \leqslant z, y \leqslant z,$ and if there is any element $Z^{*}$

Determine the value of c that makes the function f(x, y) = cxy a joint probability density function over the range 0 < x < 3 and 0 < y < x.

One of the most serious health problems in India is malaria. Consequently, Indian hospital administrators must have the resources to treat the high volume of malaria patients that are admitted. Research published in the National Journal of Community Medicine (Vol. 1, 2010) investigated whether the malaria admission rate is higher in some months than in others. In a sample of 192 hospital patients admitted in January, 32 were treated for malaria. In an independent sample of 403 patients admitted in May (five months later), 34 were treated for malaria. Describe the two populations of interest in this study.

Use a triple integral to find the volume of the following solid. The solid bounded by x = 0, x = 2, y = z, y = z + 1, z = 0 and z = 4.

Find all vectors

$$\left[\begin{array}{l} a \\ b \end{array}\right]$$

, so that the vector equation can be solved.

$$c_{1}\left[\begin{array}{r} 1 \\ -1 \end{array}\right]+c_{2}\left[\begin{array}{l} 2 \\ 1 \end{array}\right]=\left[\begin{array}{l} a \\ b \end{array}\right]$$

There are 4 blue marbles, 6 red marbles, 3 green marbles, and 2 yellow marbles in a bag. Suppose you select one marble at random. Find the probability of each outcome. Express each probability as a fraction and as a percent. Round to the nearest tenth of a percent if necessary. P(neither green nor yellow)

Compute $\varphi(n).$ $n=1795625=5^{4} \cdot 13^{2} \cdot 17$

Find the indicated partial derivatives by the method of implicit partial differentiation. $x y z+x y^{2} z^{3}-\ln z^{4}=0 ; \quad \partial z / \partial y$

Add or subtract.

$$\begin{array}{r} 8614 \\ -2030 \\ \hline \end{array}$$

Differentiate the functions. $y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$

If [K:F] is finite and u is algebraic over K, prove that [K(u):F(u)] $\leq$ [K:F].

Determine whether the statement is true or false. Use the bases of $\mathbb{R}^{2}$

$$B_{1}=\left\{\left[\begin{array}{r} 1 \\ -1 \end{array}\right],\left[\begin{array}{l} 0 \\ 2 \end{array}\right]\right\}$$

. The transition matrix from $B_{1}$ to $B_{2}$ is

$$[I]_{B_{1}}^{B_{2}}=\frac{1}{2}\left[\begin{array}{rr} -1 & 3 \\ 1 & -1 \end{array}\right]$$

.

Find an expression for the value of the following sums. $\text { a. } 2+2 \cdot 5+2 \cdot 5^{2}+\dots+2 \cdot 5^{9} \\ \text { b. } 4 \cdot 7+4 \cdot 7^{2}+4 \cdot 7^{3}+\cdots+4 \cdot 7^{12} \\ \text { c. } 1+7+13+\cdots+49 \\ \text { d. } 12+17+22+27+\cdots+92$

Determine whether the statement is true or false. If a vector space has bases S and T and the number of elements of S is n, then the number of elements of T is also n.