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
Tarskiās 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
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.
Quantifiers:
Universal Quantifier (ā): Indicates that a predicate is true for all elements within its domain.
Existential Quantifier (ā): Indicates that there is at least one element in the domain for which the predicate is true.
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.
Translation:
Converting informal language into formal quantified statements and vice versa, aiding in understanding logical expressions.
Tarskiās World:
A software program used for modeling and testing theories in first-order logic, instructional tool for visualizing and working with quantified statements.
Negating Universal and Existential Statements:
Techniques for converting universal statements into existential statements and vice versa.
Importance of understanding these transformations in logical reasoning.
Negating Universal Conditional Statements:
Identifying conditions and how to appropriately negate them to form logical equivalents.
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.
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 Ļ and ā2.
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.
Conversion:
Methods for converting rational numbers to decimal form and back, including long division.
Definition of Divisibility:
An integer a is divisible by another integer b if there exists an integer k such that a = b*k.
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.
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.
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.
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 ā¤ r < b.
Div (Division) and Mod (Modulo):
Calculation of division and modulus operations as expressions of the quotient-remainder theorem.
Absolute Value:
The absolute value of a number is its distance from zero on the number line, regardless of direction.
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.
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.
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.