To construct Venn diagrams and test the validity of the given statements and arguments, we need to first translate the statements into standard categorical forms (A, E, I, or O) and then represent them using Venn diagrams. Here's how you can approach each part:

1. Construct Venn Diagrams for Statements

a. No one is exempt from federal income tax.

• Translation to Standard Form: "No people are exempt from federal income tax."
• Subject Term: People
• Predicate Term: Exempt from federal income tax
• Standard Form: E (No S are P)

Venn Diagram:

• Draw two circles, one for "People" and one for "Exempt from federal income tax."
• Shade the area where the two circles overlap, indicating that there are no people who are exempt from federal income tax.

b. “Nothing is more useless in a developing nation’s economy than a gun.” (King Hussein I of Jordan)

• Translation to Standard Form: "No things are more useless in a developing nation’s economy than guns."
• Subject Term: Things
• Predicate Term: More useless in a developing nation’s economy than guns
• Standard Form: E (No S are P)

Venn Diagram:

• Draw two circles, one for "Things" and one for "More useless in a developing nation’s economy than guns."
• Shade the area where the two circles overlap, indicating that there are no things more useless in a developing nation’s economy than guns.

2. Draw Venn Diagrams to Test the Validity of Arguments

a. All horses are mammals, and no mammals are lizards. Therefore, no lizards are horses.

• Premise 1: All horses are mammals (A: All S are P)
• Premise 2: No mammals are lizards (E: No P are M)
• Conclusion: No lizards are horses (E: No M are S)

Venn Diagram:

• Draw three circles for "Horses," "Mammals," and "Lizards."
• Shade the area where "Horses" and "Lizards" overlap, as the conclusion states that no lizards are horses.
• The diagram should show that the conclusion logically follows from the premises.

b. Some DVDs are not film classics, but all black-and-white movies are film classics. Therefore, some black-and-white movies are not DVDs.

• Premise 1: Some DVDs are not film classics (O: Some S are not P)
• Premise 2: All black-and-white movies are film classics (A: All M are P)
• Conclusion: Some black-and-white movies are not DVDs (O: Some M are not S)

Venn Diagram:

• Draw three circles for "DVDs," "Film Classics," and "Black-and-White Movies."
• Indicate that some DVDs are outside the "Film Classics" circle.
• Show that all "Black-and-White Movies" are within the "Film Classics" circle.
• The conclusion is not necessarily valid based on the premises, as the diagram does not guarantee that some black-and-white movies are outside the "DVDs" circle.

These diagrams help visualize the logical structure of the statements and arguments, aiding in understanding their validity.

To address your request, I'll first construct Venn diagrams for each of the statements and then draw Venn diagrams to test the validity of each of the arguments. Let's start with the statements.

1. Constructing Venn Diagrams for Statements

a. No one is exempt from federal income tax.

• Subject Term: People
• Predicate Term: Exempt from federal income tax

Standard Form: ∀x (Person(x) → Not(ExemptFromFederalIncomeTax(x)))

Venn Diagram:

• Draw two circles. Label one "People" and the other "Exempt from federal income tax."
• Since no one is exempt, the "People" circle will completely overlap with the "Exempt from federal income tax" circle, indicating that there are no elements in the "Exempt from federal income tax" circle that are not also in the "People" circle.

b. “Nothing is more useless in a developing nation’s economy than a gun.” (King Hussein I of Jordan)

• Subject Term: Guns
• Predicate Term: More useless in a developing nation’s economy

Standard Form: ∀x (Gun(x) → MostUselessInDevelopingNationEconomy(x))

Venn Diagram:

• Draw two circles. Label one "Guns" and the other "More useless in a developing nation’s economy."
• Since guns are considered the most useless, the "Guns" circle will completely overlap with the "More useless in a developing nation’s economy" circle, indicating that all guns are considered the most useless.

2. Drawing Venn Diagrams to Test the Validity of Arguments

a. All horses are mammals, and no mammals are lizards. Therefore, no lizards are horses.

• Premise 1: All horses are mammals.
• Premise 2: No mammals are lizards.
• Conclusion: No lizards are horses.

Venn Diagram:

• Draw three circles. Label them "Horses," "Mammals," and "Lizards."
• Since all horses are mammals, the "Horses" circle is entirely within the "Mammals" circle.
• Since no mammals are lizards, there is no overlap between the "Mammals" and "Lizards" circles.
• Therefore, there is no overlap between the "Horses" and "Lizards" circles, which supports the conclusion that no lizards are horses. This argument is valid.

b. Some DVDs are not film classics, but all black-and-white movies are film classics. Therefore, some black-and-white movies are not DVDs.

• Premise 1: Some DVDs are not film classics.
• Premise 2: All black-and-white movies are film classics.
• Conclusion: Some black-and-white movies are not DVDs.

Venn Diagram:

• Draw two circles. Label them "DVDs" and "Film Classics."
• Since some DVDs are not film classics, there is a part of the "DVDs" circle that does not overlap with the "Film Classics" circle.
• Since all black-and-white movies are film classics, the "Black-and-White Movies" circle is entirely within the "Film Classics" circle.
• However, there is no information that directly connects "Black-and-White Movies" and "DVDs" in a way that would allow us to conclude that some black-and-white movies are not DVDs. The conclusion does not necessarily follow from the premises, making this argument invalid.

These Venn diagrams and analyses should help you understand the logical structures and validity of the given statements and arguments.