MATH 122

LaTex for assignment for Math 122. Covering topics such as:

• Logic and Proofs: Truth tables, logical equivalences, induction, and quantified reasoning.
• Set Theory: Operations, Venn diagrams, and cardinality.
• Number Theory: Modular arithmetic, gcd/lcm, and prime numbers.
• Combinatorics: Counting principles and inclusion-exclusion.
• Recursion: Fibonacci sequences and recursive proofs.

Assignment 1

Question 1

• Truth tables for logical statements.
• Logical equivalences and simplifications using De Morgan's laws.

Question 2

• Symbolic representation of statements using logical operators.
• Proof of logical equivalences with truth tables and laws of logic.

Question 3

• Proofs of equivalence for compound logical statements.
• Construction of truth tables for specific conditions.

Question 4

• Logical equivalence of compound statements.
• Use of laws like absorption and De Morgan's.

Question 5

• Symbolic reasoning with premises and conclusions.
• Validating logical arguments with truth tables and counterexamples.

Question 6 (Bonus)

• Verification of rules using logical reasoning.

---

Assignment 2

Question 1

• Boolean logic: evaluation of compound statements.
• Truth values using logical operations.

Question 2

• Translation of quantified logical statements into plain English.
• Negation of quantified statements and counterexamples.

Question 3

• Proofs using algebraic manipulation: odd and even numbers.
• Contrapositive proofs and mathematical induction.

Question 4

• Proofs of rationality and irrationality.
• Biconditional proofs involving roots and exponents.

Question 5

• Set theory: cardinality, power sets, intersections, and unions.
• Open statements and self-referential contradictions.

Question 6 (Bonus)

• Logical reasoning in constrained scenarios.

---

Assignment 3

Question 1

• Set theory and subset relationships.
• Logical equivalences involving sets and their unions/differences.

Question 2

• Logical equivalence of set operations (intersection, difference, symmetric difference).
• Proofs using Venn diagrams and set builder notation.

Question 3

• Venn diagram analysis and counterexamples to set equalities.
• Proving subset relationships between set operations.

Question 4

• Counting subsets with specific properties.
• Applications of inclusion-exclusion for problem-solving.

Question 5

• Recursively defined statements and their logical proofs.
• Negations and equivalences of recursive conditions.

---

Assignment 4

Question 1

• Mathematical induction for inequalities.
• Proofs involving summation and algebraic properties.

Question 2

• Verification of inequalities using small values and induction.
• Application of algebraic transformations to prove bounds.

Question 3

• Inductive proofs for Fibonacci sequences and summation formulas.
• Recursive patterns and their verification.

Question 4

• Induction proofs for summation identities.
• Recursive patterns for factorial-based sequences.

Question 5

• Proofs about divisors and prime factorization.
• Applications of number theory in proving the infinitude of primes.

---

Assignment 5

Question 1

• Representation of numbers in different bases.
• Conditions for logical statements about divisors and primes.

Question 2

• Greatest common divisor (gcd) and least common multiple (lcm).
• Logical proofs using gcd and lcm properties.

Question 3

• Modular arithmetic and polynomial evaluation.
• Use of remainders and modular equivalence in calculations.

Question 4

• Analysis of logical arguments using modular constraints.
• Applications of Fermat's Little Theorem in modular problems.

Question 5

• Logical reasoning with divisors and constraints.
• Counting and verifying properties of integers.