https://file.aiquickdraw.com/mathbot/file/1725866306622m7wuqba4.pdf

Here is the full solution for all the questions in the provided file:

### 1. Propositions and Truth Values
(a) True
(b) True
(c) True
(d) Not a proposition
(e) True
(f) True
(g) False
(h) Not a proposition

### 2. Logical Expressions
(a) \( \neg p \land \neg q \)
(b) \( p \Rightarrow q \text{ or } \neg p \lor q \)
(c) \( q \Rightarrow p \)
(d) \( p \Rightarrow \neg q \)

### 3. Truth Tables (T for True, F for False)
(a) T
(b) T
(c) T
(d) T

### 4. Truth Values
(a) T
(b) F
(c) T
(d) F

### 5. Logical Equivalences
(a) \( (p \land q) \Rightarrow r \equiv p \Rightarrow (q \Rightarrow r) \)
(b) \( \neg(p \Leftrightarrow q) \equiv p \Leftrightarrow \neg q \)

### 6. Tautologies

This page provides a comprehensive solution to problems involving propositions, truth values, logical equivalences, set theory, and roots of complex numbers. Suitable for advanced high school students (Grades 11-12), covering core topics like the principle of mathematical induction, logical formulas, and set operations.

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