Discrete Math - Midterm PDF

DISCREET MATHEMATICS r: "I will get wet." REVIEWER Now, we can construct some statements using these logical connectives:...

DISCREET MATHEMATICS r: "I will get wet." REVIEWER Now, we can construct some statements using these logical connectives: 1. Negation (NOT): PROPOSITIONAL LOGIC o ~p: "It is not raining." 2. Conjunction (AND): Aristotle - was the pioneer of logical reasoning. o p ∧ q: "It is raining, and I will bring an Propositional Logic is concerned with statements to umbrella." which the truth values, “true” and “false”, can be assigned. 3. Disjunction (OR): A proposition or statement is a declarative sentence that o p ∨ r: "It is raining, or I will get wet." can be classified as true or false, 4. Conditional (IF-THEN): We represent the statements by letters such as p, q and o p → q: "If it is raining, then I will bring r and we use the following symbols for and, or, and not. an umbrella." 5. Biconditional (IF AND ONLY IF): SIMPLE STATEMENTS AND COMPOUND STATEMENTS o p ↔ r: "I will get wet if and only if it is raining." Simple statement is a statement that conveys a single idea. COMPOUND STATEMENTS AND GROUPING SYMBOLS Ex: I will attend the meeting If a compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together. Compound statement is a statement is a statement that conveys two or more ideas. Symbolic Form The parenthesis indicate that: p ∧ (q ∨ ~ r ) q and ~r are grouped together. Ex: I will attend the meeting or I will go to school (p ∧ q) ∨ r p and q are grouped together. (p ∧ ~q) → (r ∨ s) p and ~q are grouped together. r and s are also grouped George Boole used symbols such as p, q, r, and s to together represent simple statements and the symbols ∧,∨,∼,→ ,and ↔ to represent connectives. If a compound statement is written as an English Logic Connectives and Symbols sentence, then a comma is used to indicate which simple statements are grouped together. Statements on Conective SForm Type the same side of a comma are grouped together. not ∼p negation and p∧q conjuction English Sentence The comma indicates that: or p∨q disjunction p, and q or not r. q and ~r are grouped together If…then p→q conditional because they are both on the If and only if p↔q biconditional same side of the comma p and q, or r. p and q are grouped together because they are both on the Example: same side of the comma Let p, q, and r represent the following. p: "It is raining." Example: q: "I will bring an umbrella." Let p, q, and r represent the following. p: "It is cold outside."  q: "I will wear a coat." F F F Example: r: "I will go for a walk." Consider the statement 2 ≤ 5 Now, let's construct some compound statements using grouping symbols. The statement's two component propositions are: Example Statement 1: 1. Proposition p : 2