Discrete Mathematics (MTH202) Lecture 1 PDF

Discrete Mathematics (MTH202) LECTURE # 1 Course Objective: 1.Express statements with the precision of formal logic 2.Analyze arguments to test their validity 3.Apply the basic properties and operations related to sets 4.Apply to sets the basic properties and ope...

Discrete Mathematics (MTH202) LECTURE # 1 Course Objective: 1.Express statements with the precision of formal logic 2.Analyze arguments to test their validity 3.Apply the basic properties and operations related to sets 4.Apply to sets the basic properties and operations related to relations and functions 5.Define terms recursively 6.Prove a formula using mathematical induction 7.Prove statements using direct and indirect methods 8.Compute probability of simple and conditional events 9.Identify and use the formulas of combinatorics in different problems 10.Illustrate the basic definitions of graph theory and properties of graphs 11.Relate each major topic in Discrete Mathematics to an application area in computing 1.Recommended Books: 1.Discrete Mathematics with Applications (second edition) by Susanna S. Epp 2.Discrete Mathematics and Its Applications (fourth edition) by Kenneth H. Rosen 1.Discrete Mathematics by Ross and Wright MAIN TOPICS: 1. Logic 2. Sets & Operations on sets 3. Relations & Their Properties 4. Functions 5. Sequences & Series 6. Recurrence Relations 7. Mathematical Induction 8. Loop Invariants 9. Loop Invariants 10. Combinatorics 11. Probability 12. Graphs and Trees Discrete Continuous Set of Integers: 3 -2 -1 0 1 2 Set of Real Numbers: Page 1 of 5 © Copyright Virtual University of Pakistan Discrete Mathematics (MTH202) -3 -2 -1 0 1 2 What is Discrete Mathematics?: Discrete Mathematics concerns processes that consist of a sequence of individual steps. LOGIC: Logic is the study of the principles and methods that distinguishes between a valid and an invalid argument. SIMPLE STATEMENT: A statement is a declarative sentence that is either true or false but not both. A statement is also referred to as a proposition Example: 2+2 = 4, It is Sunday today If a proposition is true, we say that it has a truth value of "true”. If a proposition is false, its truth value is "false". The truth values “true” and “false” are, respectively, denoted by the letters T and F. EXAMPLES: 1.Grass is green. Not Propisitions 2.4 + 2 = 6 Close the door. 2.4 + 2 = 7 x is greater than 2. 3.There are four fingers in a hand. He is very rich are propositions are not propositions. Rule: If the sentence is preceded by other sentences that make the pronoun or variable reference clear, then the sentence is a statement. Example: Example x=1 Bill Gates is an American x>2 He is very rich x > 2 is a statement with truth-value He is very rich is a statement with truth-value FALSE. TRUE. UNDERSTANDING STATEMENTS: 1.x + 2 is positive. Not a statement 2.May I come in? Not a statement 3.Logic is interesting. A statement 4.It is hot today. A statement 5.-1 > 0 A statement 6.x + y = 12 Not a statement COMPOUND STATEMENT: Simple statements could be used to build a compound statement. EXAMPLES: LOGICAL CONNECTIVES 1. “3 + 2 = 5” and “Lahore is a city in Pakistan” 2. “The grass is green” or “ It is hot today” 3. “Discrete Mathematics is not difficult to me” AND, OR, NOT are called LOGICAL CONNECTIVES. SYMBOLIC REPRESENTATION: Statements are symbolically represented by letters such as p, q, r,... EXAMPLES: Page 2 of 5 © Copyright Virtual University of Pakistan Discrete Mathematics (MTH202) p = “Islamabad is the capital of Pakistan” q = “17 is divisible by 3” CONNECTIV MEANING SYMBOL CALLED E S Negation not ~ Tilde Conjunction and ∧ Hat Disjunction or ∨ Vel Conditional if…then… → Arrow Biconditional if and only if ↔ Double arrow EXAMPLES: p = “Islamabad is the capital of Pakistan” q = “17 is divisible by 3” p ∧ q = “Islamabad is the capital of Pakistan and 17 is divisible by 3” p ∨ q = “Islamabad is the capital of Pakistan or 17 is divisible by 3” ~p = “It is not the case that Islamabad is the capital of Pakistan” or simply “Islamabad is not the capital of Pakistan” TRANSLATING FROM ENGLISH TO SYMBOLS: Let p = “It is hot”, and q = “ It is sunny” SENTENCE SYMBOLIC FORM 1.It is not hot. ~p 2.It is hot and sunny. p ∧q 3.It is hot or sunny. p∨q 4.It is not hot but sunny. ~ p ∧q 5.It is neither hot nor sunny. ~p∧~q EXAMPLE: Let h = “Zia is healthy” w = “Zia is wealthy” s = “Zia is wise” Translate the compound statements to symbolic form: 1.Zia is healthy and wealthy but not wise. (h ∧ w) ∧ (~s) 2.Zia is not wealthy but he is healthy and wise. ~w ∧ (h ∧ s) 3.Zia is neither healthy, wealthy nor wise. ~h ∧ ~w ∧ ~s Page 3 of 5 © Copyright Virtual University of Pakistan Discrete Mathematics (MTH202) TRANSLATING FROM SYMBOLS TO ENGLISH: Let m = “Ali is good in Mathematics” c = “Ali is a Computer Science student” Translate the following statement forms into plain English: 1.~ c Ali is not a Computer Science student 2.c ∨ m Ali is a Computer Science student or good in Maths. 3.m ∧ ~c Ali is good in Maths but not a Computer Science student A convenient method for analyzing a compound statement is to make a truth table for it. A truth table specifies the truth value of a compound proposition for all possible truth values of its constituent propositions. NEGATION (~): If p is a statement variable, then negation of p, “not p”, is denoted as “~p” It has opposite truth value from p i.e., if p is true, ~p is false; if p is false, ~p is true. TRUTH TABLE FOR ~p p ~p T F F T CONJUNCTION (∧ ∧): If p and q are statements, then the conjunction of p and q is “p and q”, denoted as “p ∧ q”. It is true when, and only when, both p and q are true. If either p or q is false, or if both are false, p∧q is false. TRUTH TABLE FOR p∧q p q p∧q T T T T F F F T F F F F Page 4 of 5 © Copyright Virtual University of Pakistan Discrete Mathematics (MTH202) DISJUNCTION (∨∨) or INCLUSIVE OR If p & q are statements, then the disjunction of p and q is “p or q”, denoted as “p ∨ q”.It is true when at least one of p or q is true and is false only when both p and q are false. TRUTH TABLE FOR p∨q p q p∨q T T T T F T F T T F F F Page 5 of 5 © Copyright Virtual University of Pakistan