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.