Symbolic Logic Overview PDF

SYMBOLIC LOGIC OVERVIEW 2. Inverse: ∼p → ∼q 1. Definition: 3. Contrapositive: ∼q → ∼p o Symbolic Logic: A system using Example: "She is allowed to join the volleyb...

SYMBOLIC LOGIC OVERVIEW 2. Inverse: ∼p → ∼q 1. Definition: 3. Contrapositive: ∼q → ∼p o Symbolic Logic: A system using Example: "She is allowed to join the volleyball team only if symbols and variables to represent she knows how to receive the ball." statements and logical operations.  Converse: If she knows how to receive the ball, o Statement: An assertion that is either then she is allowed to join the volleyball team. true or false.  Inverse: If she does not know how to receive the 2. Types of Statements: ball, she is not allowed to join. o Simple Statement: Does not contain other statements as parts.  Contrapositive: If she is not allowed to join, then she does not know how to receive the ball. o Compound Statement: Contains two or TRUTH TABLES more simple statements combined using logical connectives. A truth table summarizes the possible truth values of logical expressions. 3. Logical Connectives: o Negation (∼): "Not p" – Indicates the  Negation: opposite truth value of p. o p | ∼p o Conjunction (∧): "p and q" – Both p and o T|F q must be true. o F|T o Disjunction (∨): "p or q" – Either p, q, or both must be true.  Conjunction (p ∧ q): o Conditional (→): "If p, then q" – If p is o p|q|p∧q true, q must also be true. o T|T|T o Biconditional (↔): "p if and only if q" – p and q are both either true or false o T|F|F EXAMPLES OF COMPOUND STATEMENTS o F|T|F Given: o F|F|F  h: Harry is not happy.  Disjunction (p ∨ q):  v: Harry is going to watch a volleyball game. o p|q|p∨q  r: It is going to rain. o T|T|T  s: Today is Sunday. o T|F|T Write the following: o F|T|T  a: Today is Sunday and Harry is not happy → s ∧ o F|F|F ∼h  b: Today is Sunday, and Harry is not going to COMPOUND STATEMENTS AND EQUIVALENCE watch volleyball → s ∧ ∼v  Statements are equivalent if they have the same  c: If it is going to rain, then Harry is not going to truth values. watch volleyball → r → ∼v  Parentheses are used to clarify the order in which connectives are applied in compound statements. CONDITIONAL STATEMENTS AND THEIR FORMS Example Truth Table: Construct for (p ∨ q) ∧ ∼p. A conditional statement (p → q) has variations: 1. Converse: q → p PATTERNS OVERVIEW SYMMETRY 1. Definition: 1. Reflection Symmetry (Mirror Symmetry): o A pattern is a recurring, regular o A figure that can be divided into two structure or design found in nature, identical halves. human-made designs, or abstract ideas. Patterns can be modeled 2. Translation Symmetry: mathematically. o Repeated units that maintain the same o Investigating patterns helps establish figure, seen in natural structures like logical connections, make honeycombs. generalizations, predict future events, and possibly control outcomes. SEQUENCES 1. Arithmetic Sequence: KINDS OF PATTERNS o A sequence where the difference between consecutive terms is constant. 1. Patterns of Visuals: 2. Geometric Sequence: o Found in nature, like seeds, pinecones, branches, and leaves. o A sequence where the ratio between consecutive terms is constant. o Self-similar replication in plants such as 3. Harmonic Sequence: trees and ferns. 2. Patterns of Flow: o A sequence derived from the reciprocals of an arithmetic sequence. o Seen in the movement of water, the 4. Fibonacci Sequence: growth of trees, and meandering rivers. 3. Patterns of Movement: o Each term is the sum of the two preceding ones (e.g., 1, 1, 2, 3, 5, 8,...). o Rhythmic patterns in walking (left-right steps) and complex movements in THE GOLDEN RATIO animals and insects (horse walk, bird  The Fibonacci sequence is closely related to the flight). Golden Ratio (approximately 1.618), which 4. Patterns of Rhythm: appears in natural patterns, architecture, and art. o Basic patterns like heartbeats or breathing, repetitive in nature, similar to the rhythm of a body’s internal processes. 5. Patterns of Texture: o Qualities we sense through touch, such as bristly, rough, smooth, or hard surfaces. 6. Geometric Patterns: o Repeated shapes in a predictable manner, like hexagonal patterns in honeycombs or patterns on cacti and succulents.