- Illegal speeds: \(v < 45 \text{ or } v > 65\).
- X-ray frequency: \(3 \times 10^{16} \leq f \leq 3 \times 10^{19} \text{ Hz}\).
- Nonvisible light: \(f < 4 \times 10^{14} \text{ Hz} \text{ or } f > 7.5 \times 10^{14} \text{ Hz}\).
- Non-UV radiation: \(\lambda < 10 \text{ nm} \text{ or } \lambda > 400 \text{ nm}\).
- Microwave wavelength: \(1 \text{ mm} \leq \lambda \leq 1 \text{ m}\).
- Complements: 35. \(x < -3 \text{ or } x > 1\); 36. \(2 \leq x \leq 4\); 37. \(-5 < x \leq 0\); 38. \(x \leq -8 \text{ or } x > 13\).
- Conjunction/Disjunction: Conjunction is intersection (AND); Disjunction is union (OR).
Explanation
- Speed Limit (Question 30)
For the speed limit problem, the legal speed \(v\) on the highway is between \(45\) mi/hr and \(65\) mi/hr, inclusive, represented as \(45 \leq v \leq 65\). A speed is considered illegal if it falls outside of this range. Therefore, an illegal speed \(v\) is either less than \(45\) mi/hr or greater than \(65\) mi/hr. This can be expressed as a disjunction:
\[v < 45 \text{ or } v > 65\]
- Electromagnetic Spectrum (Questions 31-34)
To describe ranges of electromagnetic radiation, we use the frequencies and wavelengths typically associated with these segments of the spectrum.
- 31. Frequency of X-rays: The frequency \(f\) of X-rays typically falls in the range \(3 \times 10^{16} \text{ Hz} \leq f \leq 3 \times 10^{19} \text{ Hz}\).
- 32. Frequency of nonvisible electromagnetic radiation: Visible light has a frequency range of approximately \(4 \times 10^{14} \text{ Hz} \leq f \leq 7.5 \times 10^{14} \text{ Hz}\). Nonvisible light occurs outside this range, so: \(f < 4 \times 10^{14} \text{ Hz} \text{ or } f > 7.5 \times 10^{14} \text{ Hz}\).
- 33. Wavelength of nonultraviolet (non-UV) radiation: Ultraviolet light has a wavelength range of approximately \(10 \text{ nm} \leq \lambda \leq 400 \text{ nm}\). Non-UV radiation corresponds to wavelengths outside this range: \(\lambda < 10 \text{ nm} \text{ or } \lambda > 400 \text{ nm}\).
- 34. Wavelength of microwaves: The wavelength \(\lambda\) of microwaves is typically defined in the range \(1 \text{ mm} \leq \lambda \leq 1 \text{ m}\).
- Complementary Sets (Questions 35-38)
The complement of a set of numbers includes all values not in that set.
- 35. Original set: \(-3 \leq x \leq 1\). Complement: \(x < -3 \text{ or } x > 1\).
- 36. Original set: \(x < 2 \text{ or } x > 4\). Complement: \(2 \leq x \leq 4\).
- 37. Original set: \(x \leq -5 \text{ or } x > 0\). Complement: \(-5 < x \leq 0\).
- 38. Original set: \(-8 < x \leq 13\). Complement: \(x \leq -8 \text{ or } x > 13\).
- Conceptual Comparison (Question 39)
A conjunction of two inequality statements (using "AND") corresponds to the intersection of their solution sets, meaning the values must satisfy both conditions simultaneously. A disjunction (using "OR") corresponds to the union of their solution sets, meaning the values must satisfy at least one of the conditions. Graphically, the intersection contains only the overlap of the two regions, while the union combines the entire length covered by both regions.
Examples
Understanding these concepts is crucial for managing real-world systems, such as traffic control, engineering communications, or data filtering. For instance, in an automated factory, a sensor might trigger an alarm if a machine's temperature \(T\) exceeds a safety threshold, defined by a disjunction like \(T < 20^\circ \text{C}\) or \(T > 100^\circ \text{C}\). Similarly, engineers designing filters for signal processing use conjunctions and disjunctions to isolate specific frequency bands, such as allowing only a signal \(S\) to pass if \(100 \text{ MHz} \leq S \leq 200 \text{ MHz}\). These logical structures provide the rules for how information and resources are filtered in our daily lives.