Algebra Notes PDF

NOTES ALGEBRA https://prod-files-secure.s3.us-west-2.amazonaws.com/d5f37841-14cb- 4160-9060-cd170a3a66cd/976615fa-a9bf-4676-9b37-c640abd7419d/1- algebra.pdf Here's a structured summary of the content in Chapter 1 of the provided Algebra docu...

NOTES ALGEBRA https://prod-files-secure.s3.us-west-2.amazonaws.com/d5f37841-14cb- 4160-9060-cd170a3a66cd/976615fa-a9bf-4676-9b37-c640abd7419d/1- algebra.pdf Here's a structured summary of the content in Chapter 1 of the provided Algebra document, along with important questions and examples to explain each topic: 1. Algebraic Expressions Definition: Algebra involves combining values using operators to create expressions, where values can be constants (e.g., 1, 2) or variables (e.g., \( x \)). Constants vs. Variables: Constants are fixed values, while variables represent unknowns that can change or vary in value. Examples: Constants: \( 2 + 2 \), \( 2 \times 3 \). Variables: \( 2 \times x \), where \( x \) is unknown. Important Questions: What is the difference between a constant and a variable? Why do we use variables in algebraic expressions? How does the use of variables differ in mathematics and programming? 2. Numeric Values Types of Numbers: Natural numbers (\( N \)): Positive integers like 0, 3, 101. NOTES 1  Integers (\( Z \)): Includes negative and positive whole numbers like -20, 0. Real numbers (\( R \)): Can include fractions or decimals like -2.1, 33.4. Relationships: Every integer is a real number, and every natural number is an integer. Examples: 3 is a natural number, an integer, and a real number. 2 is an integer and a real number, but not a natural number. Important Questions: Which numeric types are suitable for representing quantities like population? Why can’t natural numbers be negative? 3. Operators Types of Operators: Includes addition (\(+\)), subtraction (\(-\)), multiplication (\(*\)), and division (\(/ \)). Inverse Operations: Addition and subtraction, multiplication and division are inverses (e.g., \( 2 + 3 = 5 \Rightarrow 5 - 3 = 2 \)). Power and Roots: Powers are repeated multiplication, and roots are the inverse (e.g., \( 2^3 = 8 \Rightarrow \sqrt{8} = 2 \)). Examples: \( 2 \times 3 = 6 \) \( 3^2 = 9 \Rightarrow \sqrt{9} = 3 \) \( 13 \div 5 = 2 \) with a remainder of \( 3 \). Important Questions: What is the relationship between multiplication and repeated addition? How does the inverse operation help solve equations? 4. Sequences Definition: A sequence is a list of numbers in a specific order. NOTES 2  Operations on Sequences: Finding the length, accessing elements, summing up elements, and extracting subsequences. Examples: \( c = [80, 34, 36, 45] \) represents the number of customers visiting a restaurant over days. Summing elements: \( \sum_{i=1}^{4} c(i) \). Important Questions: How can sequences represent real-world data like daily temperatures or sales? What is the importance of sequence length in analysis? 5. Arithmetic Operations Combining Operators: Operations like distribution (e.g., \( a \times (b + c) = a \times b + a \times c \)). Order of Operations: Parentheses take precedence, followed by powers, multiplication/division, and then addition/subtraction. Examples: \( 2 \times (3 + 4) = 14 \). Without parentheses: \( 2 \times 3 + 4 \times 5 = 26 \). Important Questions: Why is the order of operations important in evaluating expressions? How does the distributive property simplify complex calculations? 6. Equations Solving Equations: Involves finding values of variables that satisfy given conditions. Methodical Approach: Use inverse operations or trial and error to isolate the variable (e.g., \( x + 1 = 4 \Rightarrow x = 3 \)). Trial and Error: Useful for small integers (e.g., find \( x \) in \( x^3 = 27 \)). Examples: \( 2(x - 3) = 10 \Rightarrow x = 8 \). NOTES 3  \( (x + 1)(x + 2) = 12 \) can have multiple solutions. Important Questions: How do inverse operations help in solving equations? When is trial and error an efficient approach? 7. Summary of Key Concepts Types of Numbers: Natural numbers, integers, reals. Key Operators: \( +, -, *, /, **, \div, \mod \). Sequences and Summation: Understanding how sequences work and how to sum or multiply elements. Equation Solving Techniques: Using operations methodically or intuitively to find variable values. These sections cover the essentials of algebra as presented in the document, with examples and guiding questions to understand each concept better Number theory https://prod-files-secure.s3.us-west-2.amazonaws.com/d5f37841-14cb- 4160-9060-cd170a3a66cd/4f874f0c-5639-49ec-8765-638652610147/2 -numbertheory.pdf 1. Prime Numbers Definition: A prime number is a natural number greater than 1 that is not divisible evenly by any other number except 1 and itself. Testing for Primes: Use algorithms to check divisibility, such as checking factors up to the square root of the number. Sieve of Eratosthenes: An efficient method for finding all primes up to a given number by iteratively marking the multiples of each prime starting from 2. NOTES 4  Fundamental Theorem of Arithmetic: States that every natural number greater than 1 is either prime or can be uniquely written as a product of prime numbers. Examples: 997997997 is prime since it can only be expressed as 1×997. 1×9971 \times 997 90=2×3×3×590 = 2 \times 3 \times 3 \times 590=2×3×3×5. Important Questions: How do you determine if a number is prime? Why are prime numbers fundamental to number theory? How does the Sieve of Eratosthenes efficiently find primes? 2. Prime Factorization Definition: Breaking down a number into a product of its prime factors. Process: Keep dividing the number by the smallest prime until you reach 1. Examples: Prime factorization of 90: 9090 90=2×4590 = 2 \times 4590=2×45 45=3×1545 = 3 \times 1545=3×15 15=3×515 = 3 \times 515=3×5 So, 90=2×3×3×5. 90=2×3×3×590 = 2 \times 3 \times 3 \times 5 Prime factorization of 60: 6060 60=2×2×3×560 = 2 \times 2 \times 3 \times 560=2×2×3×5. Important Questions: How can you find the prime factorization of a given number? Why does the order of factorization not matter? NOTES 5  3. Large Prime Numbers Concept: There is no largest prime number. Proof Concept: Given any largest prime XXX, you can create a new number Y=1+(2×3×…×X)Y = 1 + (2 \times 3 \times \ldots \times X)Y=1+(2×3×…×X) that would either be prime or have a prime factor larger than XXX. Real-World Application: Large prime numbers are essential for encryption techniques, such as those used in secure internet communications. Important Questions: Why is it impossible for a largest prime number to exist? How are large primes used in encryption? 4. Divisors, GCD, and LCM Divisors: A divisor of a number is a smaller number that divides it evenly. Greatest Common Divisor (GCD): The largest number that divides two given numbers evenly. Finding GCD using Prime Factors: List the prime factors of each number and take the smallest count of common primes. Least Common Multiple (LCM): The smallest number that is evenly divisible by two given numbers. Finding LCM using Prime Factors: List the prime factors of each number and take the largest count of each prime. Examples: GCD of 204 and 136: 204204 136136 Prime factors of 204: [2,2,3,17]. 204204 [2,2,3,17][2, 2, 3, 17] Prime factors of 136: [2,2,2,17]. 136136 [2,2,2,17][2, 2, 2, 17] NOTES 6  Common factors: [2,2,17]. [2,2,17][2, 2, 17] GCD = 2×2×17=68. 2×2×17=682 \times 2 \times 17 = 68 LCM of 36 and 33: 3636 3333 Prime factors of 36: [2,2,3,3]. 3636 [2,2,3,3][2, 2, 3, 3] Prime factors of 33: [3,11]. 3333 [3,11][3, 11] LCM = 2×2×3×3×11=396. 2×2×3×3×11=3962 \times 2 \times 3 \times 3 \times 11 = 396 Important Questions: How can you find the GCD or LCM of two numbers using their prime factors? Why is finding the GCD important for real-world problems like arranging fence panels? 5. Applications of Number Theory Cryptography: Large primes and numbers with specific factors are crucial for encryption methods, making them fundamental to secure internet transactions. Example: Secure transmission of credit card information online relies on the difficulty of factoring large numbers into their prime components. Important Questions: How does prime factorization play a role in modern encryption? NOTES 7  Why are primes considered the building blocks of natural numbers? 6. Summary of Key Concepts Definition of Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves. Finding Primes: Techniques like prime tests and the Sieve of Eratosthenes. Prime Factorization: Expressing numbers as products of prime factors. GCD and LCM: Methods to find the greatest common divisor and the least common multiple. Importance: Applications in encryption and data security. Set Theory https://prod-files-secure.s3.us-west-2.amazonaws.com/d5f37841-14cb- 4160-9060-cd170a3a66cd/7a4f2046-9640-4aa5-8c97-f72ee416a6e6/3 -settheory_(2).pdf 1. Definition of a Set Definition: A set is an unordered collection of distinct objects, known as elements or members. Properties: Equality: Two sets are equal if they contain the same elements, regardless of order. Cardinality: The size or number of elements in a set, denoted as card(). Examples: Set of even numbers up to 10: EvensToTen = {0, 2, 4, 6, 8}. {10, 12, 14} = {14, 12, 10} because order does not matter. card({2, 3, 5}) = 3. NOTES 8  Important Questions: How do we determine if two sets are equal? What is the significance of a set’s cardinality? Why is order not important in a set? 2. Testing for Set Membership Membership Notation: p ∈ S means that p is an element of set S , and p ∉ S means it is not. Examples: 11 ∈ Evens is false, but 12 ∈ Evens is true. ”Bloggs, Fred” ∈ CSC1026 is true if "Bloggs, Fred" is part of the class list. Important Questions: What does it mean when an element is a member of a set? How can membership be tested for elements in a set? 3. Combining Sets Operations: Union ( ∪): Combines all elements from two sets. Intersection (∩): Includes only elements present in both sets. Difference (\\): Elements in the first set but not in the second. Venn Diagrams: Visual representation of set operations. Examples: A = {3, 6, 9, 12} , B = {6, 12, 18, 24}. A ∩ B = {6, 12}. A ∪ B = {3, 6, 9, 12, 18, 24}. A \\\\ B = {3, 9}. Important Questions: How can Venn diagrams help visualize set operations? What is the difference between intersection and union? NOTES 9  4. Set Comprehension Definition: Describing sets by a condition rather than listing all elements. Formal Notation: S = {x | x ∈ U · x satisfies the condition} , where U is the universal set. Combining Conditions: Use symbols like ∧ (and) or ∨ (or) to form complex conditions. Examples: {x | x ∈ {1,..., 10} · x mod 2 = 1} results in {1, 3, 5, 7, 9}. {w | w ∈ C · len(w) < 5} , where C is the set of colors, gives {”red”, ”blue”}. Important Questions: How does set comprehension simplify the definition of sets? What role does the universal set play in set comprehension? 5. Elements with Structure Sets of Pairs: A set can contain elements that are pairs, such as [x, y]. Examples: {[x, y] | x, y ∈ {0,..., 10} · x + y = 10} results in pairs that sum to 10: [[0, 10], [1, 9],..., [10, 0]]. Possible areas of picture frames with lengths: {h * v | h ∈ {20, 30}, v ∈ {30, 40}} gives {600, 800, 900, 1200}. Important Questions: How can we represent complex relationships using sets of pairs? Why is it useful to work with sets of elements that have sub- components? 6. Subsets Definition: Set A is a subset of B (A ⊆ B ) if every element in A is also in B. Proper Subset: If A is a subset of B but not equal to B , then A is a proper subset ( A ⊂ B ). Examples: NOTES 10  {1, 2} ⊆ {1, 2, 3} is true, but {1, 2, 3} ⊂ {1, 2, 3} is false. Important Questions: What is the difference between a subset and a proper subset? When is it meaningful to check if one set is a subset of another? 7. Real-World Applications and Scenarios Sports Club Example: Describing membership conditions using sets (e.g., members must register for at least one section). Penfriends Example: Constructing suitable pairs based on shared conditions like common languages or similar ages. Examples: Finding errors: Error1set = Members \\( Gym ∪ Pool ∪ Tennis). Constructing suitable penfriends: {[p1, p2] | p1, p2 ∈ P · languages(p1) ∩ languages(p2) {} ∧ abs(age(p1) − age(p2)) < 2}. Important Questions: How can set theory be used to validate data, like membership records? What are practical examples where set operations can simplify problem-solving? 8. Summary of Key Concepts Definition of a Set: Understanding the concept of an unordered collection. Set Operations: Union, intersection, and difference. Set Comprehensions: Defining sets using conditions. Structured Elements: Working with sets of pairs and subsets. Applications: Use of sets in scenarios like organizing data and validating conditions. Propositional Logic NOTES 11  https://prod-files-secure.s3.us-west-2.amazonaws.com/d5f37841-14cb- 4160-9060-cd170a3a66cd/a1235d9e-7713-4c12-b2fb-7f391e1a1e0f/4-pr opositionallogic_(1).pdf Propositional Logic: Deals with statements that are either true or false (Boolean values). Involves logical operators like AND (∧), OR (∨), and NOT (¬). Basic Logical Operators: AND (∧): True only when both statements are true. OR (∨): True if at least one statement is true. NOT (¬): Reverses the truth value of a statement. De Morgan’s Laws: ¬(A ∧ B) = ¬A ∨ ¬B ¬(A ∨ B) = ¬A ∧ ¬B Logical Deduction: Example: "If today is Tuesday, then we have a lecture." Uses chaining of rules and facts to draw conclusions (e.g., contrapositive reasoning). Truth Tables: Tables that illustrate the truth value of logical statements under various conditions. Useful for proving logical equivalences and relationships. Contrapositive: The contrapositive of A ⇒ B is ¬B ⇒ ¬A, and they are logically equivalent. NOTES 12  Chaining Logical Deductions: Combining multiple logical rules to deduce conclusions. Can be done through forward chaining (from known facts to the goal) or backward chaining (from the goal to known facts). Propositions vs. Predicates: Propositions: Statements that are either true or false (e.g., "Today is Tuesday"). Predicates: Statements with variables (e.g., "Today is X") and require additional information to be evaluated. Logical Reasoning Examples: Scenarios provided for practice involve determining conditions for truth, solving puzzles, and comparing different types of reasoning (methodical vs. lateral thinking). Logical Rules for Deduction: Standard logical rules include basic equivalences and implications (e.g., A ⇒ B). Predicate Logic Predicate Logic Overview Predicate Logic extends propositional logic by dealing with predicates and quantifiers. A predicate is a statement or function that contains variables and becomes true or false when those variables are substituted with actual values. Example: "x is greater than 3" can be written as P(x). Predicate Format: P(x) → {true, false} Key Concepts 1. Subject and Predicate: NOTES 13  Subject: The variable(s) being described. Predicate: Describes a property of the subject. 2. Examples: Let P(x) be "x is a student at QUB". Proposition: Assigning values to predicates results in a true or false proposition. E.g., P("Steve") results in a truth value. Quantifiers Universal Quantifier ( ∀): States that the predicate is true for all elements in a given domain. Example: ∀x ∈ N, P(x) means "P(x) is true for all x in the set of natural numbers." ∃ Existential Quantifier ( ): States that the predicate is true for at least one element in a given domain. Example: ∃x ∈ N, P(x) means "There exists at least one x in the set of natural numbers such that P(x) is true." Negation of Quantifiers Negating Universal Quantifier ( ∀x P(x)): ¬(∀x P(x)) becomes ∃x ¬P(x) Meaning: "There is at least one x for which P(x) is false." Negating Existential Quantifier ( ∃x P(x)): ¬(∃x P(x)) becomes ∀x ¬P(x) Meaning: "P(x) is false for all x." Operations with Quantifiers Combining Quantifiers: A predicate can involve multiple quantifiers. Example: ∃x ∈ A, ∀y ∈ B, P(x, y) reads as "There exists an x in set A such that for all y in set B, P(x, y) is true." Examples of Logical Notation NOTES 14  1. Set Example: Let P(x) = "x is even". If the domain is {1, 2, 3, 4, 5}, the truth set for P(x) is {2, 4}. 2. Negation Example: For ∀x ∈ D, P(x) where D = {1, 2, 3} , its negation is ∃x ∈ D, ¬P(x). Common Mistakes Misinterpreting Quantifiers: Careful distinction between "for all" and "there exists" is critical. Misplacing them can change the entire meaning of a logical expression. Summary of Operations Universal Quantification: ∀x ∈ D, P(x) Every x is P Existential Quantification: ∃x ∈ D, P(x) Negation: ¬(∀x ∈ D, P(x)) = ∃x ∈ D, ¬P(x) ¬(∃x ∈ D, P(x)) = ∀x ∈ D, ¬P(x) Exercise Examples 1. Prime Number Predicate: Let P(x) represent "x is prime." Universal Quantifier Example: ∀x ∈ N, P(x) states that all natural numbers are prime, which is false. 2. Negation Example: If the statement is ∀x ∈ Z, x is even , its negation would be ∃x ∈ Z, x is odd. These short notes capture key ideas in predicate logic, providing a quick revision resource during the reading week. Questions Here are the logical expressions for the provided sentences: NOTES 15 NOTES 16