Math Problem Statement

AnswerKey_Worksheet_1 (1).pdf

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Solve all questions

Solution

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) ¬p¬q\neg p \land \neg q (b) pq or ¬pqp \Rightarrow q \text{ or } \neg p \lor q (c) qpq \Rightarrow p (d) p¬qp \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) (pq)rp(qr)(p \land q) \Rightarrow r \equiv p \Rightarrow (q \Rightarrow r) (b) ¬(pq)p¬q\neg(p \Leftrightarrow q) \equiv p \Leftrightarrow \neg q

6. Tautologies

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Math Problem Analysis

Mathematical Concepts

Propositions
Truth Tables
Logical Equivalences
Set Theory
Induction
Roots of Complex Numbers

Formulas

¬(p ⇔ q) ≡ p ⇔ ¬q
(p ∧ q) ⇒ r ≡ p ⇒ (q ⇒ r)
n3 - n is divisible by 3
Cube and sixth roots of complex numbers

Theorems

Principle of Mathematical Induction
Law of Trichotomy

Suitable Grade Level

Grades 11-12