Fill in the blanks with an appropriate word, phrase, or symbol(s). To determine the validity of an argument with two premises, construct a truth table for a conditional statement of the form [(premise 1)∧(premise 2)]→_.

Fill in the blanks with an appropriate word, phrase, or symbol(s). In the decimal number 0.285714, the 7 is in the _ position.

Let

$$\begin{equation*} u=\begin{bmatrix}6\\-4\\-2\\1\\-1\end{bmatrix}\text{ and } W=\text{Span}\begin{Bmatrix} \begin{bmatrix}-9\\5\\5\\6\\-7\end{bmatrix}, \begin{bmatrix}-9\\4\\7\\2\\4\end{bmatrix}, \begin{bmatrix}4\\9\\7\\-5\\-4\end{bmatrix} \end{Bmatrix}. \end{equation*}$$

Compute the orthogonal projection matrix $P_W$, and use it to find the distance from $u$ to $W$.

Let $R$ be the region bounded by $y=x^{2}, x=1,$ and $y=0 .$ Use the shell method to find the volume of the solid generated when $R$ is revolved about the following lines.

$$y=1$$

Determine whether the statement is a tautology, self-contradiction, or neither. (~q→p)∨~q

Tell what method(s) you would use to solve the quadratic equation. Explain your choice(s).

$$m ^ { 2 } + 5 m + 6 = 0$$

Given a conditional statement p → q, find the inverse of its inverse, the inverse of its converse, and the inverse of its contrapositive.

A Van de Graaff generator is one of the original particle accelerators and can be used to accelerate charged particles like protons or electrons. You may have seen it used to make human hair stand on end or produce large sparks. One application of the Van de Graaff generator is to create X-rays by bombarding a hard metal target with the beam. Consider a beam of protons at 1.00 keV and a current of 5.00 mA produced by the generator. (a) What is the speed of the protons? (b) How many protons are produced each second?

Determine the truth value for each simple statement. Then use these truth values to determine the truth value of the compound statement. (You may have to use a reference source such as the internet.) Paris is in England or London is in France, and Berlin is in Germany.

Write an equation for each line. Then graph the line. A line whose slope and y-intercept are equal and the sum of the two is -4.

Use the fact that p→q is equivalent to ~p∨q to write an equivalent form of the given statement. If Janette buys a new car, then she sells her old car.

Use De Morgan's laws to determine whether the two statements are equivalent. (~p∨~q)→r, ~(p∧q)→r

Determine which, if any, of the three statements are equivalent. (a) The shoes are not on sale but the purse is on sale. (b) If the shoes are not on sale then the purse is not on sale. (c) It is false that the shoes are on sale or the purse is not on sale.

Use a truth table to determine whether the two statements are equivalent. q→p, ~q→~q