Chapter 1: Logic
* Defines logical operators (AND, OR, NOT) and truth tables.
* Introduces propositional logic and logical equivalence.
* Explores predicate logic and quantifiers.
Real Example: Determining the validity of a logical argument about students passing a math exam:
* Premise: All students who study hard pass the math exam.
* Premise: John studies hard.
* Conclusion: John passes the math exam.
* Logical Argument: If (study hard) then (pass exam)
John studies hard
Therefore, John passes the exam
Chapter 2: Sets
* Defines sets, set operations (union, intersection, complement), and Venn diagrams.
* Covers set theory and cardinality.
* Explores applications in counting and probability.
Real Example: Creating a set of all odd numbers less than 10:
* Set: {1, 3, 5, 7, 9}
* Operation: Union with the set of even numbers less than 10 gives {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Chapter 3: Functions
* Defines functions, domain and range, and function notation.
* Covers types of functions (injective, surjective, bijective).
* Explores applications in computer science and mathematics.
Real Example: Representing the relationship between Celsius and Fahrenheit temperature:
* Function: f(C) = (9/5)*C + 32
* Domain: Set of all Celsius temperatures
* Range: Set of all Fahrenheit temperatures
Chapter 4: Counting and Probability
* Introduces counting principles (multiplication, addition).
* Covers permutations, combinations, and tree diagrams.
* Explores probability theory and conditional probability.
Real Example: Calculating the number of different three-digit numbers that can be formed using the digits 1, 2, 3, and 4:
* Permutations: 4 x 3 x 2 = 24
* Combinations: Not applicable for this scenario
Chapter 5: Mathematical Induction
* Defines mathematical induction and its steps.
* Covers applications in proving statements about natural numbers.
* Explores recurrence relations and generating functions.
Real Example: Proving that the sum of the first n odd numbers is a perfect square:
* Base Case: n = 1 (True: 1 is a perfect square)
* Inductive Hypothesis: Assume for some k > 1 that the sum of the first k odd numbers is a perfect square.
* Inductive Step: Show that the sum of the first (k + 1) odd numbers is also a perfect square.