Let's consider the conditional statement "If it is raining, then it is cloudy" to be a true statement. a In your own words, explain how the converse statement is different from the conditional statement. b In your own words, explain how the contrapositive statement is equivalent to the conditional statement. c If we switched the hypothesis and conclusion of "All squares have four right angles" as a conditional statement, would the statement be true or false? Explain.

To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be

$$32 ^ { \circ }$$

. One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is

$$35 ^ { \circ }$$

. Estimate the height of the mountain.

Find each special product. (k+1)(k-1)

Underline the hypothesis and circle the conclusion. If two lines form vertical angles, then they intersect.

What element does thorium become if it emits an alpha particle?

The following propositions have the form $p$ or $q$. State $p$ and $q$, and give their truth value. Then determine whether the entire proposition is true or false, and explain why.

Oranges are vegetables or oranges are fruit.

Use rules of inference to show that the hypotheses “Randy works hard,” “If Randy works hard, then he is a dull boy,” and “If Randy is a dull boy, then he will not get the job” imply the conclusion “Randy will not get the job.”

Solve the inequality. Check your solution.

$$\frac{w}{6} > -3$$

Write a function that transforms

$$f ( x ) = x ^ { 4 } - 6 x ^ { 2 } – 4$$

in the of the following ways. Support your solution by using a graphing calculator. Move 3 units right, and reflect across the y-axis.

Find the argument form for the following argument and determine whether it is valid. Can we conclude that the conclusion is true if the premises are true? If George does not have eight legs, then he is not a spider. George is a spider. ∴ George has eight legs.

Explain the meaning of the given statement, which contains a multiple negation. Then answer the question that follows.
The House failed to overturn the veto on a bill that would stop logging. Based on this vote, will logging continue?

Translate these statements into English, where the domain for each variable consists of all real numbers.

a) ∀x∃y(x < y)

b) ∀x∀y(((x ≥ 0) ∧ (y ≥ 0)) → (xy ≥ 0))

c) ∀x∀y∃z(xy = z)

In each determine whether the proposed negation is correct. If it is not, write a correct negation. Statement: The product of any irrational number and any rational number is irrational. Proposed negation: The product of any irrational number and any rational number is rational.

How did each of the following help bring about the British defeat at Yorktown?