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ICMA 200 Principles and Mathematical Concepts

Course Description: Symbolic logic, proof techniques, sets, relations, functions, the real numbers, introduction to number theory.
Reference: Bridge to Abstract Mathematics: Mathematical Proof and Structures by Ronald P. Morash
Class Summary: 13 Sep 2016
  • The notion of set is a primitive, or undefined term in mathematics, analogous to point and line in plane geometry.
  • A set may be thought of as a well-defined collection of objects. The objects in the set are called elements of the set.
  • There are two methods of describing sets:
    • The roster method: for example, $A = \{2, 5, 6, 7\}$
    • The rule, or description, method: for example, $B = \{x \,|\,\text{$x$ is a prime number and $x \le 10$} \}$
  • Some special sets are $\mathbb{N}$ the set of all positive integers, $\mathbb{Z}$ the set of all integers, $\mathbb{Q}$ the set of all rational numbers, $\mathbb{R}$ the set of all real numbers, and $\mathbb{C}$ the set of all complex numbers.
  • A set $I$, all of whose elements are real numbers, is called an interval if and only if, whenever $a$ and $b$ are elements of $I$ and $c$ is a real number with $a<c<b,$ then $c\in I$.
    • Ex: Solve the following inequalities and express each solution set in interval notation:$$(\mathrm{a}) \,\, 7x - 9 \le 16 \qquad (\mathrm{b}) \,\, |2x+3| < 5 \qquad (\mathrm{c})\,\, 2x^2+x-28 \le 0$$
  • An empty, or null, set is a set with no element. We use the symbol $\{\,\,\}$ or $\emptyset$ to denote the empty set.
    • Ex: Solve the quadratic inequality $5x^2+3x+2 < 0$.
  • Homework due 15 Sep 2016:
    • Read pp. 3-23
    • Exercises p. 13, 14: 1 (b), (l), (m), 2 (a), (j), (k), (l)
Class Summary: 15 Sep 2016
  • Relations Between Sets: equality, subsets, proper subset, power set
  • Operations on Sets: Union and Intersection, Complement, Set Theoretic Difference, Symmetric Difference, Ordered Pairs and the Cartesian Product
  • Venn Diagrams
  • Algebraic Properties of Sets
    • Ex: Find $(A\cap B)\cup (A' \cap B) \cup (A\cap B')\cup (A'\cap B')$ where $A=(-\infty,4)\cup (7,\infty)$ and $B = [-2,11].$
  • Homework due 27 Sep 2016:
    • Read pp. 24-42
    • Let $M = \{y\,|\, 2x+1 \le y \le 3x-4\}$ and $N=\{y\,|\, 10\le y\le 20\}.$ What is the set of all $x$ such that $M\subset M\cap N$?
    • Let $P = \{x\,|\, y = \log_2 (1-x), y\in \mathbb{R}\}$ and $Q = \{y\,|\, y = \sqrt{x-x^2}\}.$ Find $P\cap Q.$
Class Summary: 21 Sep 2016
  • Union-Closed Sets Conjecture
    • Posed by Peter Frankl in 1979
    • A family of sets is said to be union-closed if the union of any two sets from the family remains in the family.
    • The conjecture states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family.
  • The Propositional Calculus
    • A statement, or proposition, is a declarative sentence that is either true or false, but not both true and false.
    • Compound Statements and Logical Connectives: negation (or denial), conjunction, disjunction
    • Tautology, Equivalence, the Conditional, and Biconditional
  • Homework due 27 Sep 2016:
    • Read pp. 52-63
    • Exercises p. 58, 59: 1, 3, 4
    • Exercises p. 63: 2, 3, 4
Last modified: 2016/09/22 00:07