Math 2710: Transition to Higher Mathematics
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Schedule
Chapter 3 — Counting
Counting even 5-digit numbers with digit constraints
Math-2710-UConn
Math 2710 Schedule
Chapter 1 — Sets
Sets, elements, and basic set notation
Set builder notation
Cartesian products and ordered pairs
Subsets and the subset relation
Counting subsets of finite sets (the 2^n formula)
Element membership vs. subset notation
Power sets
Set operations: union, intersection, and difference
Set complement with respect to a universal set
Venn diagrams
Indexed families of sets
Chapter 2 — Logic
Statements and truth values
Logical connectives: and, or, not
Conditional statements and implication
Biconditionals and logical equivalence
Truth tables
Logical equivalence and De Morgan’s laws
Quantifiers: universal and existential
Paired quantifiers
Implications with quantified open sentences
Translating English into logical notation
Negations of quantified and conditional statements
Deduction rules: modus ponens, proof by cases
Chapter 3 — Counting
The multiplication principle for counting ordered lists
Multiplication principle applied to Cartesian products
Counting binary strings
The addition principle and disjoint unions
Counting even 5-digit numbers with digit constraints
Permutations and k-permutations
Combinations and binomial coefficients C(n,k)
Pascal’s triangle and the binomial theorem
Chapter 4 — Direct Proof
Theorems, definitions, propositions, and axioms
The division algorithm and the fundamental theorem of arithmetic
Direct proof: assuming the hypothesis and deriving the conclusion
AM-GM inequality and proof techniques
The role of definitions in proof construction
Least common multiple and interpreting definitions
Proof by cases and the triangle inequality
Chapter 5 — Proof by Contrapositive and Congruences
Proof by contrapositive
Congruence modulo n and properties of congruent integers
Arithmetic operations on congruences
Chapter 6 — Proof by Contradiction
Proof by contradiction (reductio ad absurdum)
Euclid’s proof of infinitely many primes
Chapter 7 — Other Proof Techniques
If-and-only-if proofs (bidirectional implication)
Existence proofs: constructive vs. non-constructive
Euclid’s algorithm for the GCD
Uniqueness proofs
Further constructive and non-constructive existence proofs
Euclidean algorithm via division and subtraction
Chapter 8 — Proofs about Sets
Proving set membership using element characterization
Set inclusion proofs (A ⊆ B)
Set equality proofs (showing A = B via mutual inclusion)
Chapter 9 — Disproof
Disproving conjectures by counterexample
Disproving universal statements; historical conjectures
Disproving existence claims via universal negation
Chapter 10 — Mathematical Induction
Introduction to mathematical induction
Induction proofs: sums, divisibility, and algebraic identities
Advanced induction examples including the harmonic series
Strong induction and its relationship to weak induction
Strong induction applications including divisibility
Fundamental theorem of arithmetic: prime factorization uniqueness
Fibonacci numbers and the golden ratio
Well-ordering principle and minimal counterexamples
Strong induction: non-intersecting segments connecting red and blue points
Chapter 11 — Relations
Relations as generalizations of equality and ordering
Properties of relations: reflexive, symmetric, transitive
Equivalence relations and equivalence classes
Problems on equivalence relations
Partitions and equivalence relations
Integers modulo n as equivalence classes
Relations between different sets
Chapter 12 — Functions
Functions as special relations; domain and codomain
Equality of functions
Problems distinguishing functions from relations
Injective, surjective, and bijective functions
Proving injectivity, surjectivity, and bijectivity
The pigeonhole principle
Function composition
Inverse relations and inverse functions
When the inverse relation is itself a function
Image and preimage of sets under functions
Proving bijectivity of a specific cubic rational function
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Examples
Chapter 3 — Counting
Counting even 5-digit numbers with digit constraints
Examples
(Example 3.6) How many even 5 digit numbers are there for which:
no digit is zero
the digit
\(6\)
appears exactly once.