CSC 133 Test 3 Review

3.1
For Quantified Statements: Define predicate, domain, truth set
Universal and existential quantifier and notation
Identifying true & false quantified statements
Translation between informal & formal language for quantified statements
Tarski’s World
3.2
Negating Universal and Existential statements
Negating Universal Conditional Statements
4.3
Fully understand what a Rational (and Irrational) number is and how it related to Integers
Convert Rationals to Decimals and back
4.4
What is Divisibility?
What is a prime number?
How do Divisibility and Prime relate?
Factoring and the Unique Factorization of Integers (also, the Standard factored form)
4.5
Quotient-Remainder theorem: what are the parts?
Div & Mod: How to calculate them
What is the absolute value of a number?
4.6
What are Floor and Ceiling for various numbers?
Using Floor and Ceiling in calculations
Using Floor and Ceiling to Calculate Div and Mod

3.1 Quantified Statements

  1. Definitions:

    • Predicate: A function that returns true or false based on the values of its variables.

    • Domain: The set of all possible inputs for the variables in a predicate.

    • Truth Set: The subset of the domain for which the predicate is true.

  2. Quantifiers:

    • Universal Quantifier (∀): Indicates that a predicate is true for all elements within its domain.

    • Existential Quantifier (∃): Indicates that there is at least one element in the domain for which the predicate is true.

  3. Identifying True & False Quantified Statements:

    • Techniques for evaluating the validity of quantified statements.

    • Examples to illustrate the differences between true and false statements for both quantifiers.

  4. Translation:

    • Converting informal language into formal quantified statements and vice versa, aiding in understanding logical expressions.

  5. Tarski’s World:

    • A software program used for modeling and testing theories in first-order logic, instructional tool for visualizing and working with quantified statements.

3.2 Negation of Quantified Statements

  1. Negating Universal and Existential Statements:

    • Techniques for converting universal statements into existential statements and vice versa.

    • Importance of understanding these transformations in logical reasoning.

  2. Negating Universal Conditional Statements:

    • Identifying conditions and how to appropriately negate them to form logical equivalents.

4.3 Rational and Irrational Numbers

  1. Rational Numbers:

    • Defined as numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

    • Examples include fractions and integers, as integers can be written as a fraction.

  2. Irrational Numbers:

    • Defined as numbers that cannot be represented as a simple fraction, and their decimal representation is non-repeating and non-terminating.

    • Common examples include π and √2.

  3. Relationship with Integers:

    • Rational and irrational numbers are part of the broader set of real numbers, which also includes integers.

    • Understanding the distinctions between these sets.

  4. Conversion:

    • Methods for converting rational numbers to decimal form and back, including long division.

4.4 Divisibility

  1. Definition of Divisibility:

    • An integer a is divisible by another integer b if there exists an integer k such that a = b*k.

  2. Prime Numbers:

    • Defined as numbers greater than 1 that have no positive divisors other than 1 and themselves.

    • Examples include 2, 3, 5, and 7.

  3. Relationship of Divisibility and Prime:

    • Prime numbers play a crucial role in the fundamental theorem of arithmetic (unique factorization theorem), where every integer greater than 1 can be uniquely factored into prime numbers.

  4. Factoring and Unique Factorization:

    • Understanding the process of factoring numbers into their prime components.

    • Expressing integers in standard factored form facilitates solving problems related to divisibility.

4.5 Quotient-Remainder Theorem

  1. The Parts of the Theorem:

    • For any integers a and b (b > 0), there exist unique integers q (the quotient) and r (the remainder) such that a = b*q + r, where 0 ≤ r < b.

  2. Div (Division) and Mod (Modulo):

    • Calculation of division and modulus operations as expressions of the quotient-remainder theorem.

  3. Absolute Value:

    • The absolute value of a number is its distance from zero on the number line, regardless of direction.

4.6 Floor and Ceiling Functions

  1. Definitions:

    • Floor: The greatest integer less than or equal to a given number.

    • Ceiling: The smallest integer greater than or equal to a given number.

  2. Using Floor and Ceiling in Calculations:

    • Application of these concepts in calculations, especially for real-number approximations and rounding functions.

    • Examples of how floor and ceiling functions are used in mathematical expressions.

  3. Calculating Div and Mod Using Floor and Ceiling:

    • Showcasing how to apply floor and ceiling functions when computing division and modulus.

These concepts are foundational for understanding more complex mathematical theories and for application in various real-world scenarios.

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